Question

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

Answer 1

`sum_(n=0)^5 0.1(4)^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:

`0.4/0.1=4`

`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>=0`.

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

`0.1(4^n)=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(4)^n`:

`sum_(n=0)^5 0.1(4)^n`