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Chapter 8: Problem 54

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$a+a r+a r^{2}+\dots+a r^{n-1}$$

Short Answer

Expert verified
The sum \(a+a r+a r^{2}+\dots+a r^{n-1}\) can be expressed in summation notation as \(\sum_{i=1}^{n} a r^{i-1}\).

Step by step solution

01

Recognize the geometric sequence

Recognize the pattern of the given sequence. It's geometric with a common ratio of \(r\), and the first term is \(a\).
02

Express the sequence terms

In a geometric sequence, the \(nth\) term is given by \(a r^{n-1}\), where \(n\) is the term number, starting from 1.
03

Express in summation notation

To express this series in summation notation, note that the sum is over the terms from \(a r^{0}\) to \(a r^{n-1}\). This sum can be represented in summation notation as \(\sum_{i=1}^{n} a r^{i-1}\), where \(i\) is the index of summation and corresponds to the term number in the sequence.

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