Find the sum of an infinite geometric series if the first term and the common ratio are both -1/3. Common Ratio. The common ratio r=2. 2) Student/Teacher Actions (what students and teachers should be doing to facilitate learning) 1. For example. The sequence 9,3,1,1/3,… = is a geometric sequence with common ratio 1/3. So a1 and a3 can take the values (-9/2 , -2) , (9/2 , 2). The number of terms must be positive integer, the first term can be in terms of real numbers or variables, and the common ratio must be nonzero real number. Python Program to find Sum of Geometric Progression Series Example. A recursive formula for the sequence 100, 80, 64, 51. Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Identifying Geometric Sequences Tell whether each sequence is geometric. You can find it by dividing two consecutive pairs of terms. So we will need to use the formula for the last term of an arithmetic. The sum of the numbers in a geometric progression is also known as a geometric series. The fact that a geometric sequence has a common factor allows you to do two things. asked by Eric on January 17, 2019. You can put this solution on YOUR website! find the common ratio of an infinite geometric series with first term 6 and with sum 7. The series is not geometric because it fails to have a common ratio. And each time I'm multiplying it by a common number, and that number is often called the common ratio. General form of geometric progression :. Find each sum. Because consecutive terms of a geometric sequence change by equal factors, the points of any geometric sequence with a positive common ratio lie on an exponential curve. This video shows you how to find the common ratio of a geometric sequence by dividing the second term by the first term. 3 - Geometric Sequences. Suppose a term of a geometric sequence is a4 = 121. The only two series that have methods for which we can calculate their sums are geometric and telescoping. 8 , r = −5 16) a 1 = 1, r = 2 Given the first term and the common ratio of a geometric sequence find the recursive formula and the three terms in the sequence after the last one given. The terms of the series are frequently fractions. * To find the common ratio of every geometric sequence, divide a pair of terms. Find which term of the arithmetic progression will next be equal to a term of the geometric progression. It doesn't matter which pair as long as they're right next to each other. is geometric, because each step divides by 3. The first term is a 1 , the common ratio is r, and the number of terms is n. Solution This is an arithmetic series, because the diﬀerence between the terms is a constant value, 2·5. The fourth term is 10 and the seventh term is 80 Find the common ratio. 5, 1, 7, 3, 9, « 62/87,21 There is no common difference. Therefore, the sum of the first 10 terms of the geometric series. how do i find the common ratio of the infinite geometric ratio series with the given sum and first term? for example: sum=4 first term=7. , moving from term to term) give rise to equal changes in the output (determined by the common difference). You should be able to recognise something. Example: Given the geometric sequence 2 , 4 , 8 , 16 ,. A geometric series with 7 terms that begins with 500 and successively decreases by 10%. For example, the series 2, 6, 18, 54,. A geometric progression “GP” consists of a sequence of numbers such that the value obtained by the division of the (n+1) th term by the n th term is the same as that obtained when the n th term divided by the (n – 1) th term. It would depend on the data given. ind the common ratio and write out the first four terms of the geometric sequence {3^n-1/6} Common ratio is a1=, a2=, a3=, a4= The answer: A geometric sequence has the (general) form: a_n = a_1 * (r)^(n - 1) a_n = a with a subscript of n (this is the nth term in the sequence) a_1 = a with a subscript of 1 (this is the 1st term in the sequence). A recursive formula for the sequence 81, 27, 9, 3, « is a1 = 81, an = an± 1, n 2. This Python program allows the user to enter first value, total number of items in a series, and the common ration. Let's verify the nth term of our piggy bank problem:. (C) The sequence is arithmetic with common difference 0 and it is also geometric with common ratio 1. Algebra 2/Trig: Chapter 6 - Sequences and Series In this unit, we will… Identify an arithmetic or geometric sequence and find the formula for its nth term Determine the common difference in an arithmetic sequence Determine the common ratio in a geometric sequence. so, 16 = divide both sides by 2 ==> 16/3 =. If so, find the common ratio. Find a rule for the nth term. Find the sum of an infinite geometric series if the first term and the common ratio are both -1/3. First I substituted 121. Both sequences have 1 as their first term. To determine what the ratio is, we must look at the current term to the previous term. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. Geometric Series A pure geometric series or geometric progression is one where the ratio, r, between successive terms is a constant. Menu Algebra 2 / Sequences and series / Geometric sequences and series A geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio, r. Find the Sum of the Infinite Geometric Series This is a geometric sequence since there is a common ratio between each term. Next we will utilize the General Formulas for both the Arithmetic and Geometric Sequences and use them to find any term in the sequence, as well as the nth term. The first three terms of a geometric sequence are 4, 16, and 64. RE: Find the common ratio of the geometric sequence need help? 1. Let's verify the nth term of our piggy bank problem:. Geometric series: the expression formed by adding the terms of a finite geometric sequence. , This represents the first term of a geometric sequence? , This represents the number of terms in a geometric sequence? , The variable 'r' represents this in a geometric sequence? Vocabulary Find the Next 3 Terms. an represents the nth term, the unknown term that you are trying to find, of a sequence. If we start with a particular first term, and then multiply the same number successively, we obtain a geometric sequence. Anyway, have a look at the following:-The sequence upto the 6th term is as follows:-1/3, -1/9 ,1/27, -1/81, 1/243, -1/729 I got these two extra terms just by using the previous term and mutiplying it by the common ratio as follows:-1st term = 1/3. A geometric sequence has an initial value of 6 and a common ratio of 2. Identify the common difference or common ratio. To find the sum of the first Sn terms of a geometric sequence use the formula. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term. With inputs from experts, these worksheets are tailor-made for high-school students. Geometric Sequence In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. In this case, we are given the first and fourth terms: a n = a 1 r n − 1 U s e n = 4. 15) a 1 = 0. 9) a 1 = −1, r = −2 Find a 11 10) a 1 = −4, r = 3. The terms of the series are frequently fractions. It doesn't matter which pair as long as they're right next to each other. Find the next four The first term of a geometric sequence is -3, and the common ratio is 3 2 terms. So, fourth term = here first term is given as 2and fourth term is 16. For example, the geometric sequence { 1, 3, 9, 27 } has a common ratio of 3. To find the common ratio, divide any term by its preceding term. For example, the series 2, 6, 18, 54,. There is a common ratio of 7KHVHTXHQFHLVJHRPHWULF Use the formula for a geometric sequence. A geometric sequence is a series of numbers where each number is found by multiplying the previous number by a constant. $16:(5 Determine whether each sequence is arithmetic, geometric, or neither. consecutive terms of a geometric sequence is called the common ratio. Negative, the terms will alternate between positive and negative. To define an arithmetic or geometric sequence, we have to know not just the common difference or ratio, but also the initial value (called a). Find the first term and the common ratio. It doesn't matter which pair is chosen as long. Thus, if we know the first two terms of a geometric sequence, then we can find the equation for the nth term. Find the common ratio of the geometric sequence -3, 6, -12, 24,. As you know, the terms in an arithmetic progression go: Next, factorise the LHS. 216, 72, 24, 8. Since the ratios are constant, the sequence is geometric. This ratio is usually indicated by the variable r. Then find the next two terms. Thus, if we know the first two terms of a geometric sequence, then we can find the equation for the nth term. This constant ratio is called the common ratio and is denoted by r. Find the first term. Find all terms between a 1 = − 5 and a 4 = − 135 of a geometric sequence. In this sequence: a is the first term and r the common ratio. Find a 12 Given a term in a geometric sequence and the common ratio find the term named in the problem, the explicit formula, and the three terms in the sequence after the last one given. The sequences are either arithmetic or geometric. a1 is the first term in a sequence. Example: Find the sum of the first six terms of the geometric sequence with first term −3and common ratio 4. The proof is similar to the one used for real series, and we leave it for you to do. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. MrRicciardi73 38,872 views. Numbers are said to be in Geometric Sequence if there is a common ratio between any two consecutive terms. what is the common ratio in the sequence?. [3] For example, if you wish to find the 8th term in the sequence, then n = 8. CHAPTER 12: SEQUENCES, PROBABILITY, AND STATISTICS 711 This means the easy way to recognize a geometric sequence is just to divide several pairs of consecutive terms and see if you get the same number every time. This relationship allows for the representation of a geometric series using only two terms, r and a. You can discover more about the geometric series below the tool. Example 1: Write the first five terms of the geometric sequence whose first term is 3 and whose ration is 2. Find the 1st term, the common ratio and the sum of the first 10 terms. 15) a1 = 0. The height of the bounces shown in the table above form a geometric sequence. P to infinity or for a number of terms would be given. 02 while r = 0. Find the common ratio of the following geometric sequence. Example 1: Find the common ratio. Since the common ratio has value between -1 and 1, we know the series will converge to some value. Engaging math & science practice! Improve your skills with free problems in 'Find the common ratio of the geometric sequence' and thousands of other practice lessons. Ask students to find the patterns. • Use geometric sequences to model and solve real-life problems. It would depend on the data given. Geometric Sequences. an represents the nth term, the unknown term that you are trying to find, of a sequence. The sequence 9,3,1,1/3,… = is a geometric sequence with common ratio 1/3. A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. For example, the sequence 2, 6, 18, 54, is a geometric progression with common ratio 3. In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. We would need to know a few terms so that we could calculate the common ratio and ultimately the formula for the general term. Sequences and Series. In a previous video, we derived the formula for the sum of a finite geometric series where a is the first term and r is our common ratio. Formula for nth term a = —6 and r = 2 b. [1]Step 2, Calculate the common ratio (r) of the sequence. Finding the Terms of a Geometric Sequence:. To do so, we would need to know two things. 1 1 1 n n ar S r We must first find the common ratio, r, and the first term in the geometric sequence a 1 5 3 2 2 2 53 4 51 5 1 1 1 1 8 1 8 4 1 1 1 4 16 16 4 2 2 Do you see why we were told that the common ratio was positive? 11 4 4 64 2 16 64 1 1 1 12 64 1 256 1 1 1 2 n n. It doesn't matter which pair is chosen as long. Find the 8th term in this geometric sequence. Solved examples to find the Sum of first n terms of the Geometric Progression: 1. As used in my blog post above, but applied to your question, R = 1. Use r to find the geometric means. Where, a is the first term and r is the common ratio. The common ratio (r) is obtained by dividing any term by the preceding term, i. Write rules for geometric sequences. The terms of the series are frequently fractions. A geometric sequence increase or decrease by a common factor - the common ratio. Example (d). Solution This is an arithmetic series, because the diﬀerence between the terms is a constant value, 2·5. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. Example 2: Find the common ratio if the fourth term in geometric series is$\frac{4}{3}$and the seventh term is$\frac{64}{243}\$. Geometric Series A pure geometric series or geometric progression is one where the ratio, r, between successive terms is a constant. Precalculus Sequences & Series Test Practice Name_____ Sequence Formulas: a n = a 1 + d (n - 1) 1 1 n a a r n Series Formulas : 1 (1 ) 1 n n ar S r Determine if the sequence is arithmetic or geometric. The nth term of a geometric sequence is , where is the first term and is the common ratio. This Site Might Help You. A Geometric series is a series with a constant ratio between successive terms. The formula for the nth term of a geometric sequence is given by An = ar^(n - 1), where a is the first term, n is the term number and r is the common ratio. List the first four terms and the 10th term of a geometric sequence with a first term of 3 and a. the first term is given by a 1 = 5 and the common ratio is r = 0. This result would be the same number that gets us from the second to the third term. 216, 72, 24, 8. You can input integers, decimals or fractions. find the common difference. , This represents the first term of a geometric sequence? , This represents the number of terms in a geometric sequence? , The variable 'r' represents this in a geometric sequence? Vocabulary Find the Next 3 Terms. Numbers are said to be in Geometric Sequence if there is a common ratio between any two consecutive terms. Browse through the geometric sequence worksheets consist of exercises in finding the first term and common ratio, appending the next few terms, finding the general term of the geometric sequence, word problems and more. These ratios are not the same, and therefore, by definition, these three terms do not have a common ratio, because "common ratio" literally means "the same ratio. ” The behaviour of a geometric sequence depends on the value of the common ratio. In order to find the common ratio in a geometric sequence, take a term and divide it by the term before it. 3 - Geometric Sequences. So, what if we want to find this sum? Luckily, there's a formula! To find the sum of the first n terms of a geometric sequence: Let's use it:. R and r are different. The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (that is, if you find the ratio of) successive terms, you'll always get this common value. 13) a 2 = 12, r = -3 Find a 9 14) a 5 = -64, r = -2 Find a 10 Given two terms in a geometric sequence find the common ratio, the term named in the. The common ratio r=2. There are two ways of finding the common ratio of a geometric sequence: (1) The first one is to divide the number and the number after it. With geometric sequences, we'll multiply by something each time. This ratio is called _____. so general term of geometric sequence is ar n-1. ) Find the sum of the first ten terms of the geometric series 16 – 48 + 144 – 432 + … (6. Geometric Sequence In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. Geometric Sequences. Key Concept Geometric Sequence A geometric sequence with a starting value a and a common ratio r is a sequence of the form a, ar, ar2, ar3, A recursive definition for the sequence has two parts: initial condition an an_l r, for n > I. Browse through the geometric sequence worksheets consist of exercises in finding the first term and common ratio, appending the next few terms, finding the general term of the geometric sequence, word problems and more. The common ratio is the number multiplied to get the next term. If so, find the common. Next we will utilize the General Formulas for both the Arithmetic and Geometric Sequences and use them to find any term in the sequence, as well as the nth term. If so, identify the. And each time I'm multiplying it by a common number, and that number is often called the common ratio. View and Download PowerPoint Presentations on Finding The Nth Term Of A Sequence PPT. ) The first term of a geometric sequence is 2 1 and r = 3 2. The formula for the nth term of a geometric sequence is given by An = ar^(n - 1), where a is the first term, n is the term number and r is the common ratio. Geometric Series. Quite common in maths books. Finding Common Ratios. 1 (1 ); infinite: r a S. [2][3] A common ratio is also called a geometric ratio. Plugging into the summation formula, I get:. find an exponential function that passes through the points of a geometric sequence learn about half-life for exponential decay and doubling time for exponential growth In Chapter 1, you used recursive formulas to model geometric growth and decay. Graph the sequence. (Total 6 marks) 6. Find the common ratio of the sequence - 172 - 86 - 43 - Sign In. Click here to get an answer to your question - The second term in a geometric sequence is 50. The constant factor between consecutive terms of a geometric sequence is called the common ratio. (D) The sequence is arithmetic with common difference 21. 15) a1 = 0. asked • 05/28/15 Find the common ratio of an infinite geometric series with the given sum and first term. For example, the sequence 2, 4, 8, 16, … 2, 4, 8, 16, \dots 2, 4, 8, 1 6, … is a geometric sequence with common ratio 2 2 2. A sequence a 1, a 2, a 3, ,a n is said to be geometric is the ratio between consecutive terms remains. Sn=a1(1−rn)1−r,r≠1, where n is the number of terms, a1 is the first term and r is the common ratio. If the series is geometric, find the common ratio. Find the common ratio of the geometric sequence 32,8,2,1/2 plss answer this question - 2309595. Find common ratio and write out the first four terms of the geometric sequence {bn} = {(5/2)^n} The answer: A geometric sequence has the (general) form: b_n = b_1 * (r)^(n - 1) b_n = b with a subscript of n (this is the nth term in the sequence) b_1 = a with a subscript of 1 (this is the 1st term in the sequence) n = number of terms r = the. Similar to an arithmetic sequence, a geometric sequence is determined completely by the first term a, and the common ratio r. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson Geometric sequence. Geometric sequence, b. Geometric Sequences. ind the common ratio and write out the first four terms of the geometric sequence {3^n-1/6} Common ratio is a1=, a2=, a3=, a4= The answer: A geometric sequence has the (general) form: a_n = a_1 * (r)^(n - 1) a_n = a with a subscript of n (this is the nth term in the sequence) a_1 = a with a subscript of 1 (this is the 1st term in the sequence). 5th term of 3,—,— 4 16' 64 6. We can obtain the common ratio by. Formula for Mh. You can find it by dividing two consecutive pairs of terms. Problem 3 Find the scale factor and the command ratio of a geometric progression if a 5 - a 1 = 15 a 4 - a 2 = 6 Solution: there are two geometric progressions. 𝑎 𝑛=(− 2 3) 𝑛 Solution: a. So if someone were to tell you, hey, you've got a geometric sequence. List the first four terms and of a geometric sequence with a first term of 2 and a common ratio of. Since this ratio is common to all consecutive pairs of terms, it is called the common ratio. Choose the answer from the following : Determine the 5th term of the geometric sequence {image} Choose the answer from the following : The common ratio in a geometric sequence is {image} , and the fourth term is {image}. 1 (1 ); infinite: r a S. Now write the sum of the 4th and 6th terms and factorise. A geometric sequence, or geometric progression, is a sequence of numbers where each successive number is the product of the previous number and some constant r. These worksheets introduce the concepts of arithmetic and geometric series. Corn too n Vs Find the fifth term and the nth term of the geometric sequence whose initial term a and common ratio r are given. Find the common ratio of the geometric sequence -3, 6, -12, 24,. Is this a geometric sequence? 3. My question. The fixed number is called common ratio. Real life problem with geometric sequences. Find a 11 Given the first term and the common ratio of a geometric sequence find the term named in the problem, the explicit formula, and the recursive formula. Let's figure out a rule that gets us from one number to the next. In a number sequence, order of the sequence is important, and depending on the sequence, it is possible for the same terms to appear multiple times. Find each sum. Find The First Term And The Common Ratio For The Given Sequence. Which is the RECURSIVE rule for finding the Nth term of a geometric sequence? A) nth term is just the term before it to the power of the common ratio B) nth term is the first term plus the common ratio to the (n+1) power C) nth term is just the term before it times the common ratio D) nth term is the following term raised to the nth power E. an = a1 + (n – 1)d– This is the formula. Common Ratio. The first term a1 is 100, and n 2. A geometric series has the form "a*r^k", where "a" is the first term of the series, "r" is the common ratio and "k" is a variable. Now I'll give some examples of geometric sequences. Create a real-life situation in which a geometric series would be used. Find sums of fi nite geometric series. In a geometric sequence aa 35 16 4. Another way of saying this is that each term can be found by multiplying the previous term by a certain number. 1 to 5 is 1 x 5. Find the common ratio and write out the first 4 terms of the given geometric sequence. 13) a 2 = 12, r = -3 Find a 9 14) a 5 = -64, r = -2 Find a 10 Given two terms in a geometric sequence find the common ratio, the term named in the. The proof is similar to the one used for real series, and we leave it for you to do. A geometric series that starts with 4, and ends with ­8748, and has common ratio of ­3. a1 is equal to 90 and your common ratio is equal to negative 1/3. Taking a pair of numbers from the sequence and dividing them produces the common ratio, providing the numbers chosen border each other. The first term of a geometric progression exceeds the second term by $$2$$, and the sum of the second and third terms is $$\frac{4}{3}$$. Physical Science A ball is dropped from a height of 500 meters. Example 4: Find the 8th term, if the first term and the common ratio of a geometric sequence are 45 and 0. 12) Gabe and Erik are finding the 9 th term of the geometric sequence -5, 10, -20, …. P Series Sn = a(r n) / (1- r) Tn = ar (n-1). Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson Geometric sequence. Write a rule for the nth term. In this sequence: a is the first term and r the common ratio. Name Class Date. The term r is the common ratio, and a is the first term of the series. a Find the first term of the series. b) Show work and explain which option Lidia should choose. How to find the first term and common ratio if given that sum of first and third term is 20 and sum of fourth and sixth term is 540? Relevant page. Consequently, you can multiply any term in the sequence by the common ratio to obtain the next term of the sequence. [3] For example, if you wish to find the 8th term in the sequence, then n = 8. CCSS MODELING A landscaper is building a brick patio. Find the indicated term of the geometric sequence. Find the next four The first term of a geometric sequence is -3, and the common ratio is 3 2 terms. These worksheets introduce the concepts of arithmetic and geometric series. Also, a geometric sequence has p as its common ratio. Tutorial on geometric sequences and summations. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term. If the series is geometric, find the common ratio. called the common ratio, such that. Arithmetic and Geometric Sequences Practice Homework For each Sequence, Pattern, Table, or Story below identify whether it is Arithmetic or Geometric, find the common difference or common ratio, write an Explicit Formula, then use your formulas to find the given. Identifying Geometric Sequences Tell whether each sequence is geometric. A geometric sequence is a sequence in which the ratio consecutive terms is constant. Tap for more steps Simplify the numerator. The yearly salary values described form a geometric sequence because they change by a constant factor each year. Dhelmalyz P. Let's verify the nth term of our piggy bank problem:. For example, in the sequence below, the common ratio is 2, because each term is 2 times the term before it. This is a geometric sequence since there is a common ratio between each term. Geometric sequences of numbers. Find the sum of the first 20 terms to the nearest whole number. MrRicciardi73 38,872 views. To find the common ratio , find the ratio between a term and the term preceding it. This Python program allows the user to enter first value, total number of items in a series, and the common ration. Find the sum of the geometric series: 4 - 12 + 36 - 108 + to 10 terms. 3: Geometric Sequences and Series - Mathematics LibreTexts. Is the following sequence Arithmetic, Geometric or Neither? If so, what is the common difference or common ratio? 506, 403, 300, 197, 94, What is an Arithmetic Sequence with a common difference of -103?. Show that the sequence is geometric. The yearly salary values described form a geometric sequence because they change by a constant factor each year. ” The behaviour of a geometric sequence depends on the value of the common ratio. Let's figure out a rule that gets us from one number to the next. Because consecutive terms of a geometric sequence change by equal factors, the points of any geometric sequence with a positive common ratio lie on an exponential curve. If the series is geometric, find the common ratio. In the following examples, the common ratio is found by dividing the second term by the first term, a 2 /a 1. Since this ratio is common to all consecutive pairs of terms, it is called the common ratio. Geometric Series is a sequence of terms in which next element is obtained by multiplying common ration to previous element. let us substitute: Then we have two values for the common difference: Then the sequesnces are: As this is a geometric series we have a common ratio between the consecutive terms. (B) The sequence is geometric with common ratio 23, but it is not arithmetic. What is the common ratio of the G. [1] 7) A population of ants is growing at a rate of 8% a year. Let's do the sum of the first few terms:. Thus, the common ratio is 1/5. Geometric sequences; 2. Let's do the sum of the first few terms:. 5 for an, 4 for n, and 3 for r in the general form. Determine whether the sequence 6, 18, 54, 162 is geometric. *10 Points*? More questions Find the common ratio for the following geometric sequence 70, 7, 0. ” The behaviour of a geometric sequence depends on the value of the common ratio. 62/87,21 Subtract each term from the term that follows it. Write the formula for this sequence in the form an = a1 ⋅ rn−1. 3 - Geometric Sequences. • Use geometric sequences to model and solve real-life problems. Identifying Geometric Sequences In a geometric sequence, the ratio of any term to the previous term is constant. In a number sequence, order of the sequence is important, and depending on the sequence, it is possible for the same terms to appear multiple times. Then give a recursive definition and a closed formula for the number of dots in the $$n$$th pattern. (c) Find the sum of the infinite sequence. You have a pattern in your sequence. Find the common ratio of the following geometric sequence. Find all terms between a 1 = − 5 and a 4 = − 135 of a geometric sequence. The yearly salary values described form a geometric sequence because they change by a constant factor each year. 𝑎 𝑛=(− 2 3) 𝑛 Solution: a. Find the common ratio of the geometric sequence -3, 6, -12, 24,.