Problem 46 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$5+5^{2}+5^{3}+\dots+5^{12}$$

Step-by-Step Solution

Answer

The sum of the series \(5+5^{2}+5^{3}+\dots+5^{12}\) in summation notation is \(\sum_{i=1}^{12} 5.5^{i-1}\).

Step 1: Identify the pattern

Look at the series: \(5+5^{2}+5^{3}+\dots+5^{12}\). Clearly, we can observe a pattern here. It's a geometric series where each term is 5 raised to an increasing power. The index 'i' can be seen as going from 1 to 12.

Step 2: Use the summed geometric series formula

The series can be expressed in sigma notation, which is used to denote a sum. The general finding for a geometric series \(a+ar+ar^{2}+ar^{3}+\dots+ar^{n}\) with 'a' as the first term and 'r' as the common ratio is given in the sigma notation by \(\sum_{i=1}^{n} ar^{i-1}\). Here \(a=5\), and \(r=5\), as 5 is both the first term and the common ratio.

Step 3: Form the summation notation

Substituting the values of \(a\) and \(r\) into the sigma notation we get \(\sum_{i=1}^{12} 5.5^{i-1}\). This expression represents the sum of the series from \(5^1\) to \(5^{12}\).