Question

Question #c18f2

Answer 1

`7` terms in total:
`27+9+3+1+1/3+1/9+1/27=1093/27`

Explanation:

The series is a geometric series and is thus in the form `a_n=a_(n-1)*r` or `a_n=a_1*r^(n-1)`.

The first term, `a_1`, is `27`. The common ratio `r` is `1/3`.

We need to find `n` such that `sum_(k=1)^n27*(1/3)^(k-1)=1093/27`.

The sum of a geometric series is equal to `(a_1(1-r^(n-1)))/(1-r)`. Thus, the left-hand side can be expressed as `(27(1-(1/3)^(n-1)))/(1-1/3)=81/2(1-1/3^(n-1))`.

Solve `81/2(1-1/3^(n-1))=1093/27`. This becomes `1/3^(n-1)=1-2186/2187=1/2187`. Solving for `n` results in `n=7`.