SOLUTION: The sum to infinity of a geometric series is 4 times the second term. A) find the common ratio B) the first term of the series is 32. What is the percentage error in the approxim

Algebra ->  Sequences-and-series -> SOLUTION: The sum to infinity of a geometric series is 4 times the second term. A) find the common ratio B) the first term of the series is 32. What is the percentage error in the approxim      Log On


   

Question 935078: The sum to infinity of a geometric series is 4 times the second term.
A) find the common ratio
B) the first term of the series is 32. What is the percentage error in the approximation of S5=S?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Let the first term be b, and the common ratio be r .
Term number n is b%5Bn%5D=b%2Ar%5E%28n-1%29 ---> b%5B2%5D=b%2Ar
The sum of the first n terms is

and it only converges to a number (the sum to infinity),
S=b%2F%281-r%29 , if and only if r%3C1 .
"The sum to infinity of a geometric series is 4 times the second term" translates as
b%2F%281-r%29=4%2A%28b%2Ar%29-->1%2F%281-r%29=4%2Ar-->1%2F4=r%281-r%29-->1%2F4=r-r%5E2-->r%5E2-r%2B1%2F4=0-->%28r-1%2F2%29%5E2=0-->highlight%28r=1%2F2%29
If the first term is b=32 , then

and S=32%2F%281-1%2F2%29=32%2F%281%2F2%29=32%2A%282%2F1%29=64
If we calculate S=64 as an approximation for S%5B5%5D=62 ,
the absolute error is 64-62=2 .
As a percentage of the true value of S%5B5%5D=62 ,
that error is the relative error, 2%2F62=1%2F31=(approx.)0.03226=highlight%28%223.226%25%22%29