`sum_(n = 1)^8 5(-5/2)^(n - 1)` Find the sum of the finite geometric series.

The geometric series is:

`sum_(n=1)^8 5(-5/2)^(n-1)`

The given summation notation is in the form

`sum_(n=1)^n a_1 *r^(n-1)`

It can be seen that the first term and common ratio of the geometric sequence are a1=5 and r=-5/2.Plugging in these two values to the formula of finite geometric series

`S_n=a_1 *(1-r^n)/(1-r)`

then, the sum of the first eight terms is:

`S_8=5 * (1-(-5/2)^8)/(1-(-5/2))=-278835/128`

Therefore,  `sum_(n=1)^8 5 (-5/2)^(n-1) = -278835/128` .