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Chapter 14: Problem 99

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
To find the sum of the first \(n\) terms of a geometric sequence without manually adding all the terms, apply the geometric sum formula \( S_n = a(1 - r^n)/(1 - r)\) by substituting the values of the first term \(a\), the common ratio \(r\), and the number of terms \(n\) into the formula, then simplify.

Step by step solution

01

Identify the parameters of the sequence

Identify the first term \(a\), the common ratio \(r\), and the number of terms \(n\) in the given geometric sequence.
02

Apply the Sum formula to the sequence

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

Simplify the expression

Perform the operations as indicated in the formula. Calculate the power, perform the multiplication and carry out the division. The result is the sum of the first \(n\) terms of the geometric sequence.

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Most popular questions from this chapter

Solve: \(2 x^{2}=4-x\).

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{n}{n+1} ; n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$

Use the formula for the sum of an infinite geometric series to solve Exercises. A new factory in a small town has an annual payroll of \(\$ 6\) 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?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}$$

Use the formula for the general term (the nth term) of a geometric sequence to solve. Suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the thirtieth day of the month?
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