SOLUTION: The first 2 terms of a geometric series have a sum of -4. The fourth and fifth terms have a sum of 256. Find the first term and the common ratio of the series.

Algebra ->  Sequences-and-series -> SOLUTION: The first 2 terms of a geometric series have a sum of -4. The fourth and fifth terms have a sum of 256. Find the first term and the common ratio of the series.      Log On


   

Question 726593: The first 2 terms of a geometric series have a sum of -4. The fourth and fifth terms have a sum of 256. Find the first term and the common ratio of the series.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
In a geometric series each term equals the previous one times the common ratio:
b%5Bn%2B1%5D=b%5Bn%5D%2Ar (recursive formula)
so that all terms can be calculated as a function of the first term as
b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29

In this problem,
b%5B1%5D%2Bb%5B2%5D=-4
b%5B4%5D%2Bb%5B5%5D=256
and we want to find b%5B1%5D and r

MY WAY:
b%5B4%5D=b%5B1%5D%2Ar%5E3
b%5B5%5D=b%5B2%5D%2Ar%5E3
b%5B4%5D%2Bb%5B5%5D=b%5B1%5D%2Ar%5E3%2Bb%5B2%5D%2Ar%5E3
256=%28b%5B1%5D%2Bb%5B2%5D%29%2Ar%5E3
256=%28-4%29%2Ar%5E3
r%5E3=256%2F%28-4%29
r%5E3=-%282%5E8%2F2%5E2%29
r%5E3=-2%5E6 --> r=-2%5E2 --> highlight%28r=-4%29
b%5B1%5D%2Bb%5B1%5D%2Ar=b%5B1%5D%2Bb%5B2%5D
b%5B1%5D-4b%5B1%5D=-4
-3b%5B1%5D=-4
b%5B1%5D=-4%2F%28-3%29 --> highlight%28b%5B1%5D=4%2F3%29

ANOTHER WAY:
b%5B2%5D=b%5B1%5D%2Ar
b%5B1%5D%2Bb%5B2%5D=b%5B1%5D%2Bb%5B1%5D%2Ar
b%5B1%5D%2Bb%5B2%5D=b%5B1%5D%281%2Br%29
b%5B1%5D%281%2Br%29=-4
b%5B4%5D=b%5B1%5D%2Ar%5E3
b%5B5%5D=b%5B1%5D%2Ar%5E4
b%5B4%5D%2Bb%5B5%5D=b%5B1%5D%2Ar%5E3%2Bb%5B1%5D%2Ar%5E4
b%5B4%5D%2Bb%5B5%5D=b%5B1%5D%2A%28r%5E3%2Br%5E4%29
b%5B1%5D%2A%28r%5E3%2Br%5E4%29=256
system%28b%5B1%5D%281%2Br%29=-4%2Cb%5B1%5D%2A%28r%5E3%2Br%5E4%29=256%29 --> b%5B1%5D%2A%28r%5E3%2Br%5E4%29%2F%28b%5B1%5D%281%2Br%29%29=256%2F%28-4%29 --> %28r%5E3%2Br%5E4%29%2F%281%2Br%29=-64 --> r%5E3%281%2Br%29%2F%281%2Br%29=-64 --> r%5E3=-64 --> r=root%283%2C-64%29 --> highlight%28r=-4%29
system%28b%5B1%5D%281%2Br%29=-4%2Cr=-4%29 --> b%5B1%5D%281-4%29=-4 --> b%5B1%5D%281-4%29=-4 --> b%5B1%5D=-4%2F%28-3%29 --> highlight%28b%5B1%5D=4%2F3%29