SOLUTION: Write the following geometric series in summation notation -9/2 + 3/2 - 1/2. + 1/6 - ...+. 1/39366 Using the formula for the sum of a geometric series, compute the sum.

Algebra ->  Sequences-and-series -> SOLUTION: Write the following geometric series in summation notation -9/2 + 3/2 - 1/2. + 1/6 - ...+. 1/39366 Using the formula for the sum of a geometric series, compute the sum.       Log On


   

Question 1097575: Write the following geometric series in summation notation
-9/2 + 3/2 - 1/2. + 1/6 - ...+. 1/39366
Using the formula for the sum of a geometric series, compute the sum.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
39366=2%2A3%5E9 , so 1%2F39366=1%2F%282%2A3%5E9%29=3%5E%28-9%29%2F2 .
The series is
.
It is a geometric series with 12 terms,
first term b=-9%2F2 and common ratio r=-1%2F3 .
The sum of the first n terms of a geometric series with
first term b and common ratio r is
S%5Bn%5D=b%28r%5En-1%29%2F%28r-1%29 .
In this case, blindly applying that formula, we would calculate


A SMARTER WAY:
If we pair those 12 terms, we have 6 terms
%28-9%2F2%2B3%2F2%29%2B%28-1%2F2%2B1%2F6%29%2B%22...%22=-3%2B-1%2F3%2B%22...%22
forming a geometric series with first term -3 and ratio 1%2F9
The sum then can be calculated as
=-3%2A728%2A730%2F%288%2A9%5E5%29=-3%2A91%2A730%2F9%5E5=-91%2A730%2F3%5E9=-66430%2F19683