SOLUTION: I need to find the sum of the first n terms of the g.p. 4,12,36... and find the least value of n for which the sum is greater than 500,000. I know that r = 3 and a = 4 but then

Algebra ->  Sequences-and-series -> SOLUTION: I need to find the sum of the first n terms of the g.p. 4,12,36... and find the least value of n for which the sum is greater than 500,000. I know that r = 3 and a = 4 but then       Log On


   

Question 328501: I need to find the sum of the first n terms of the g.p. 4,12,36... and find the least value of n for which the sum is greater than 500,000.
I know that
r = 3 and a = 4 but then I am stuck.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
the formula is:
S+=+a%2A%28%281-r%5E%28n%2B1%29%29%2F%281+-+r%29%29
a+=+4
r+=+3
S+=+4%2A%28%281-3%5E%28n%2B1%29%29%2F%281+-+3%29%29
S+=+4%2A%28%281-3%5E%28n%2B1%29%29%2F%28-2%29%29
I'll find out when the sum is greater than 500000
S+=+4%2A%281-3%5E%28n%2B1%29%29%2F%28-2%29
S+=+%28-2%29%2A%281-3%5E%28n%2B1%29%29
S%2F%28-2%29+=+1+-+3%5E%28n%2B1%29
Multiply both sides by -1
S%2F2+%2B+1+=+3%5E%28n%2B1%29
I can take the log of both sides
log%28S%2F2+%2B+1%29+=+%28n%2B1%29%2Alog%283%29
If S+=+50000, the left side is close to log%282.5%29+%2B+4, so
%28log%282.5%29+%2B+4%29%2Flog%283%29+=+n+%2B+1
n+=+%28log%282.5%29+%2B+4%29%2Flog%283%29+-+1
n+=+4.39794%2F.477121+-+1
n+=+9.21766+-+1
n+=+8.21766
Putting this back into formula,
S+=+4%2A%28%281-3%5E%28n%2B1%29%29%2F%28-2%29%29
S+=+4%2A%28%281-3%5E9.21766%29%2F%28-2%29%29
S+=+4%2A%281+-+25000.08621%29%2F%28-2%29
S+=+-24999.08612%2A%28-2%29
S+=+49998.17225
The next whole number for n is 9
I think that would put S over 50000
8 seems to be too small