Skip to main content
Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 29
Chapter 9, Problem 29

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...

Verified step by step guidance
1
Identify the first term \( a_1 \) of the geometric sequence. Here, the first term is \( -\frac{3}{2} \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{3}{-\frac{3}{2}} \).
Recall the formula for the sum of the first \( n \) terms of a geometric sequence: \[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Substitute \( a_1 = -\frac{3}{2} \), \( r \) (from step 2), and \( n = 14 \) into the sum formula.
Simplify the expression step-by-step to find the sum of the first 14 terms, being careful with signs and powers of \( r \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence -3/2, 3, -6, 12, ..., each term is multiplied by -2 to get the next term.
Recommended video:
Guided course
4:18
Geometric Sequences - Recursive Formula

Common Ratio

The common ratio in a geometric sequence is the fixed factor between consecutive terms. It is found by dividing any term by the previous term. Identifying the common ratio is essential for applying the sum formula and understanding the sequence's behavior.
Recommended video:
5:57
Graphs of Common Functions

Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. This formula helps find the total of the sequence's terms efficiently.
Recommended video:
Guided course
4:18
Geometric Sequences - Recursive Formula
Related Practice
Textbook Question

Use the Fundamental Counting Principle to solve Exercises 29–40. A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

4
views
Textbook Question

Use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

Textbook Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. a1=-20, d = -4

3
views
Textbook Question

Use mathematical induction to prove that each statement is true for every positive integer n. ∑i=1n5⋅6i=6(6n−1)\(\sum\)_{i=1}^{n} 5 \(\cdot\) 6^i = 6(6^n - 1)

3
views
Textbook Question

Use the Fundamental Counting Principle to solve Exercises 29–40. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or sport). In how many ways can you order the car?

Textbook Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)6