# How do you use summation notation to expression the sum 0.1+0.4+1.6+...+102.4?

Apr 15, 2018

${\sum}_{n = 0}^{5} 0.1 {\left(4\right)}^{n}$

#### Explanation:

Judging by the first three terms, they don't share a common difference, so it will not be an arithmetic series; however, we should test the ratio of the terms to determine whether we have a geometric series:

$\frac{0.4}{0.1} = 4$

$\frac{1.6}{0.4} = 4$

So, we have a geometric series with the first term $a = 0.1 ,$ and common ratio $r = 4.$ This means that the $n$th term in the series is the first term $0.1$ multiplied by ${4}^{n} , n \ge 0$.

Since the series ends at $102.4 ,$ and we want to determine how many $n$ we have, we'll solve this:

$0.1 \left({4}^{n}\right) = 102.4$

${4}^{n} = 1024$

$n = 5$

So, the series starts at $n = 0 ,$ ends at $n = 5 ,$ and the terms are given by $0.1 {\left(4\right)}^{n}$:

${\sum}_{n = 0}^{5} 0.1 {\left(4\right)}^{n}$