Title

College Algebra
Tutorial 54D: Geometric Sequences and
Series

 Since a geometric sequence is a sequence, you find the terms exactly the same way that you do a sequence.   n is our term number and we plug the term number into the function to find the value of the term. If you need a review on sequences, feel free to go to Tutorial 54A: Sequences. Lets see what we get for our first five terms:

 *n = 1

 *n = 2

 *n = 3

 *n = 4

 *n = 5

 What would be the common ratio for this problem? If you said -5 you are correct!  Note that you would have to multiply (-10)(-5) = 50, (50)(-5) = -250, (-250)(5) = 1250, and (1250)(-5) = -6250.   It has to be consistent throughout the sequence.  Also note that the base that is being raised to a power is -5.

 Since a geometric sequence is a sequence, you find the terms exactly the same way that you do a sequence.   n is our term number and we plug the term number into the function to find the value of the term. If you need a review on sequences, feel free to go to Tutorial 54A: Sequences. Lets see what we get for our first five terms:

 *n = 1

 *n = 2

 *n = 3

 *n = 4

 *n = 5

 What would be the common ratio for this problem? If you said 1/4 you are correct!  Note that you would have to multiply by 1/4 each time you go from one term to the next: (4)(1/4) = 1, (1)(1/4) = 1/4,  (1/4)(1/4) = 1/16, and (1/16)(1/4)=1/64.   It has to be consistent throughout the sequence.  Also note that the base that is being raised to a power is 1/4.

 Answer/Discussion to 2a 125, 25, 5, 1, ...

 We will use the nth term formula for a geometric sequence,  to help us with this problem. Basically we need to find two things: the first term of the sequence,  and  the common ratio, r. What is , the first term? If you said 125, give yourself a high five.  The first term of this sequence is 125. What is r, the common ratio? If you said 1/5, you are right!! Note that you would have to multiply 1/5 each time you go from one term to the next: (125)(1/5) = 25, (25)(1/5) = 5, and (5)(1/5) = 1.   It has to be consistent throughout the sequence. Putting in 125 for and 1/5 for r we get:

 Answer/Discussion to 2b 4, -12, 36, -108, ...

 We will use the nth term formula for a geometric sequence,  to help us with this problem. Basically we need to find two things: the first term of the sequence,  and  the common ratio, r. What is , the first term? If you said 4, give yourself a high five.  The first term of this sequence is 4. What is r, the common ratio? If you said -3, you are right!! Note that you would have to multiply -3 each time you go from one term to the next: (4)(-3) = -12, (-12)(-3) = 36, and (36)(-3) = -108.   It has to be consistent throughout the sequence. Putting in 4 for and -3 for r we get:

 Answer/Discussion to 3a 2 + 14 + 98 + 686 + 4802 + 33614 + 235298

 We will use the formula for the sum of the first n terms of geometric sequence,  , to help us with this problem. Basically we need to find three things: the first term of the sequence, the common ratio, and how many terms of the sequence we are adding in the series. What is , the first term? If you said 2 you are right! What is r, the common ratio? If you said 7, give yourself a pat on the back. Note that you would have to multiply -2 each time you go from one term to the next: (2)(7) = 14, (14)(7) = 98, (98)(7) = 686, (686)(7) = 4802, (4802)(7) = 33614, and (33614)(7) = 235298.   It has to be consistent throughout the sequence. How many terms are we summing up? If you said 7, you are correct. Putting in 2 for the first term, 7 for the common ratio, and 7 for n, we get:

 We will use the formula for the sum of the first n terms of geometric sequence,  , to help us with this problem. Basically we need to find three things: the first term of the sequence, the common ratio, and how many terms of the sequence we are adding in the series. What is , the first term? If you said 12.25 you are right!   Since this summation starts at 2, you need to plug in 2 into the given formula: What is r, the common ratio? If you said -3.5, give yourself a pat on the back.  Note that -3.5 is the number that is being raised to the exponent.  So each time the number goes up on the exponent, in essence you are multiplying it by -3.5. How many terms are we summing up? If you said 9, you are correct.  If you start at 2 and go all the way to 10, there will be 9 terms. Putting in 12.25 for the first term, -3.5 for the common ratio, and 9 for n, we get:

 We will use the formula for the sum of infinite geometric sequence, ,  to help us with this problem. Basically we need to find two things: the first term of the sequence and the common ratio. What is the first term, ? If you said -1 you are right!   Since this summation starts at 0, you need to plug in 0 into the given formula: What is the common ratio, r? If you said -.1, give yourself a pat on the back. Note that -.1 is the number that is being raised to the exponent.  So each time the number goes up on the exponent, in essence you are multiplying it by -.1. Putting in -1 for the first term and -.1 for the common ratio we get: