Problem 54 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(a\).

Step 2: 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.

Step 3: 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.