Question

Evaluate and round to the nearest hundredth

Answer 1

`5.33`

Explanation:

`\sum_(1)^(10)4\left((1)/(4)\right)^(n-1)=4\sum_(1)^(10)\left((1)/(4)\right)^(n-1)\\ \\=4\left[\left((1)/(4)\right)^0+\left((1)/(4)\right)^1+\left((1)/(4)\right)^2+\dots+\left((1)/(4)\right)^9\right]\\ \\=4\left[1+(1)/(4)+\left((1)/(4)\right)^2+\dots+\left((1)/(4)\right)^9\right]`

`1+(1)/(4)+\dots+\left((1)/(4)\right)^9` is a geometric series with first element
`(a)=1`, constant ratio
`(r)=(1)/(4)` and total number
`(n)=10`

sum of
`n` terms of geometric series
`=(a(1-r^(n)))/(1-r)`

`1+(1)/(4)+\dots+\left((1)/(4)\right)^9=(1\left(1-\left((1)/(4)\right)^(10)\right))/(\left(1-(1)/(4)\right))}\\ \\=1.3333\\ \\4\left[1+(1)/(4)+\dots+\left((1)/(4)\right)^9\right]\\ \\=4* 1.3333\\=5.33\\ \\\sum_(1)^(10)4\left((1)/(4)\right)^(n-1)=5.33`