Question

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given. $$ a_1=1 ; \quad r=-\frac{1}{3} $$

Name: Problem 24, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-16T22:17:01-08:00
Duration: 1 min 47 s
Channel: Dale Sanford
Description: Find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. a1=1 ; r=-(1)/(3)

   Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=1 ; \quad r=-\frac{1}{3}
$$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 11, Problem 24

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Solved by Expert Dale Sanford

06/16/2020

Step 1

The nth term $ a_n $ of a geometric sequence can be found using the formula: \[ a_n = a_1 \cdot r^{n-1} \] where $ a_1 $ is the first term and $ r $ is the common ratio. Show more…

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Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given. $$ a_1=1 ; \quad r=-\frac{1}{3} $$

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Key Concepts

Geometric Sequence

A geometric sequence is an ordered list of numbers where each term after the first is obtained by multiplying the preceding term by a constant factor. This constant factor is known as the common ratio, and the sequence is defined by its first term and the common ratio.

Common Ratio

The common ratio is the constant factor that each term of a geometric sequence is multiplied by to obtain the next term. It determines the rate at which the sequence grows or decays and is crucial in determining the behavior and formula for any term in the sequence.

nth Term Formula for a Geometric Sequence

The nth term of a geometric sequence can be calculated using the formula a_n = a1 × r^(n?1), where a1 is the first term and r is the common ratio. This formula provides a method to directly compute any term in the sequence without needing to calculate all preceding terms.

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