Question

What is the sum of the series sum from n equals 1 to infinity of negative 7 times three eighths to the n power question mark.

Answer 1

`-(21)/(5)`

Explanation:

The sum of the series ∑ from n=1 to infinity of -7(3/8)^n is a geometric series with first term a = -7(3/8)^1 = -21/8 and common ratio r = 3/8. The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by:

∑
`_(n=1)^(\infty) ar^(n-1) = (a)/(1-r)`

Substituting the values of a and r, we get:

∑
`_(n=1)^(\infty) -7\left((3)/(8)\right)^n = (-(21)/(8))/(1-(3)/(8)) = (-(21)/(8))/((5)/(8)) = \boxed{-(21)/(5)}`

Answer 2

-21/5

Explanation:

The given series is:-7(3/8)^1 - 7(3/8)^2 - 7(3/8)^3 - ...

We can see that this is a geometric series with first term a = -7(3/8)^1 = -21/8 and common ratio r = 3/8.

The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by:

sum = a/(1-r)

Substituting the values for a and r, we get:sum = (-21/8)/(1-3/8) = (-21/8)/(5/8) = -21/5Therefore, the sum of the given series is -21/5.