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Chapter 11: Problem 91

Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.

Short Answer

Expert verified
Without adding all terms, the sum of the first \(n\) terms of a geometric sequence can be found by substituting the terms into the formula \(S_n = a (1 - r^n) / (1 - r)\) and calculating the result.

Step by step solution

01

Identify the parameters of the geometric sequence

Identify the first term \(a\) of the sequence, the common ratio \(r\), and the number of terms \(n\) which you want to find the sum of.
02

Substitute into the formula

Substitute the values for \(a\), \(r\) and \(n\) into the formula \(S_n = a (1 - r^n) / (1 - r)\).
03

Solve the equation

After substitution of the values, solve the equation to get the sum of the first \(n\) terms of the geometric sequence.

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Most popular questions from this chapter

You are dealt one card from a 52-card deck. Find the probability that you are dealt a 7 or a red card.

What is the common ratio in a geometric sequence?

A new factory in a small town has an annual payroll of S6 million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. $$ \begin{array}{cccc} {16,} & {0.96(16),} & {(0.96)^{2}(16),} & {(0.96)^{3}(16)} \\ {\text { Ist }} & {2 \text { nd }} & {3 \text { rd }} & {4 \text { th }} \\ {\text { swing }} & {\text { swing }} & {\text { swing }} & {\text { swing }} \end{array} $$ After 10 swings, what is the total length of the distance the pendulum has swung?

What is a geometric sequence? Give an example with your explanation.
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