SOLUTION: the sum of the first term of an exponential sequence is 135 and the sum of the third and the fourth term is 60 given that the common ratio is positive calculate the limit of the su

Algebra ->  Sequences-and-series -> SOLUTION: the sum of the first term of an exponential sequence is 135 and the sum of the third and the fourth term is 60 given that the common ratio is positive calculate the limit of the su      Log On


   

Question 1181525: the sum of the first term of an exponential sequence is 135 and the sum of the third and the fourth term is 60 given that the common ratio is positive calculate the limit of the sum of the first n^th term as n become large
Found 2 solutions by mananth, greenestamps:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
There appears to be typo

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Presumably the first statement is supposed to be....

the sum of the cross%28first%29 cross%28term%29 first and second terms of an exponential sequence is 135.

Then the problem has a "nice" solution.

Let a and r be the first term and common ratio. Then

a%2Bar=135
ar%5E2%2Bar%5E3+=+r%5E2%28a%2Bar%29+=+r%5E2%28135%29+=+60

r%5E2+=+60%2F135+=+4%2F9

Since we are told r is positive, r is 2/3. Then

a%2Bar+=+a%2B%282%2F3%29a+=+%285%2F3%29a+=+135
a+=+%283%2F5%29%28135%29+=+81

The infinite sum is first term divided by (1 minus the common ratio):

a%2F%281-r%29+=+81%2F%281-2%2F3%29+=+81%2F%281%2F3%29+=+81%2A3+=+243

ANSWER: 243