Respuesta :
A geometric series is written as [tex] ar^n [/tex], where [tex] a [/tex] is the first term of the series and [tex] r [/tex] is the common ratio.
In other words, to compute the next term in the series you have to multiply the previous one by [tex] r [/tex].
Since we know that the first time is 6 (but we don't know the common ratio), the first terms are
[tex] 6, 6r, 6r^2, 6r^3, 6r^4, 6r^5, \ldots [/tex].
Let's use the other information, since the last term is [tex] 4374 > 6 [/tex], we know that [tex] r>1 [/tex], otherwise the terms would be bigger and bigger.
The information about the sum tells us that
[tex] \displaystyle \sum_{i=0}^n 6r^i = 6\sum_{i=0}^n r^i = 6558 [/tex]
We have a formula to compute the sum of the powers of a certain variable, namely
[tex] \displaystyle \sum_{i=0}^n r^i = \cfrac{r^{n+1}-1}{r-1} [/tex]
So, the equation becomes
[tex] 6\cfrac{r^{n+1}-1}{r-1} = 6558 [/tex]
The only integer solution to this expression is [tex] n=6, r=3 [/tex].
If you want to check the result, we have
[tex] 6+6*3+6*3^2+6*3^3+6*3^4+6*3^5+6*3^6 = 6558 [/tex]
and the last term is
[tex] 6*3^6 = 4374 [/tex]
