Asked by Miracle
Find the sum of the nth terms of a G.P 5+15+45+...........What is the smallest number of the terms which will give a total of more than 10^8.
Answers
Answered by
Steve
a = 5
r = 3
Sn = a(1-r^n)/(1-r) = 5(3^n - 1)/2
So, we want
5(3^n-1)/2 > 10^8
3^n-1 > 4*10^7
3^n > 4*10^7 - 1
n > log(4*10^7 - 1)/log3
Now, using base 10 logs, and ignoring the useless -1,
n > (
n > 7+log4)/log3
n > 15.9
So the first 16 terms will sum to more than 10^8
As a sanity check, 5*3^15 = 7*10^7 so I figure just the last two or three terms will produce the desired amount, and the first 13 terms are just noise.
r = 3
Sn = a(1-r^n)/(1-r) = 5(3^n - 1)/2
So, we want
5(3^n-1)/2 > 10^8
3^n-1 > 4*10^7
3^n > 4*10^7 - 1
n > log(4*10^7 - 1)/log3
Now, using base 10 logs, and ignoring the useless -1,
n > (
n > 7+log4)/log3
n > 15.9
So the first 16 terms will sum to more than 10^8
As a sanity check, 5*3^15 = 7*10^7 so I figure just the last two or three terms will produce the desired amount, and the first 13 terms are just noise.
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.