Question

The first term of a finite geometric series is 6 and the last term is 4374. The sum of all the term os 6558. find the common ratio and calculate the number of terms in the series.

Answer 1

A geometric series is written as
`ar^n`, where
`a` is the first term of the series and
`r` is the common ratio.

In other words, to compute the next term in the series you have to multiply the previous one by
`r`.

Since we know that the first time is 6 (but we don't know the common ratio), the first terms are

`6, 6r, 6r^2, 6r^3, 6r^4, 6r^5, \ldots`.

Let's use the other information, since the last term is
`4374 > 6`, we know that
`r>1`, otherwise the terms would be bigger and bigger.

The information about the sum tells us that

`\displaystyle \sum_(i=0)^n 6r^i = 6\sum_(i=0)^n r^i = 6558`

We have a formula to compute the sum of the powers of a certain variable, namely

`\displaystyle \sum_(i=0)^n r^i = \cfrac{r^(n+1)-1}{r-1}`

So, the equation becomes

`6\cfrac{r^(n+1)-1}{r-1} = 6558`

The only integer solution to this expression is
`n=6, r=3`.

If you want to check the result, we have

`6+6*3+6*3^2+6*3^3+6*3^4+6*3^5+6*3^6 = 6558`

and the last term is

`6*3^6 = 4374`