Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 27

Chapter 9, Problem 27

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 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...

Verified step by step guidance

Identify the first term $ a_1 $ of the geometric sequence. In this sequence, the first term is $ 3 $.

Determine the common ratio $ r $ by dividing the second term by the first term: $ r = \frac{-6}{3} $.

Recall the formula for the sum of the first $ n $ terms of a geometric sequence: \[ S_n = a_1 \times \frac{1 - r^n}{1 - r} \] where $ a_1 $ is the first term, $ r $ is the common ratio, and $ n $ is the number of terms.

Substitute the values $ a_1 = 3 $, $ r $ (from step 2), and $ n = 11 $ into the formula: \[ S_{11} = 3 \times \frac{1 - r^{11}}{1 - r} \].

Simplify the expression by calculating $ r^{11} $, then perform the subtraction and division inside the fraction, and finally multiply by 3 to find the sum of the first 11 terms.

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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, -6, 12, -24, ..., the common ratio is -2.

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 applies when r is not equal to 1.

Substitution and Simplification

To find the sum, substitute the known values (a = 3, r = -2, n = 11) into the sum formula and simplify carefully. This involves calculating powers of the ratio and performing arithmetic operations to get the final sum.

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