math studies
posted by ll on .
An arithmetic series has a first term of 4 and a common difference of 1. A geometric series has a first term of 8 and a common ratio of 0.5. After how many terms does the sum of the arithmetic series exceed the sum of the geometric series?

anyone????

please

can anyone help?

for the AS
S_{n} = n/2(8 + n1)
= n/2(n9)
for a GS
S_{n} = 8(1.5^n)/(1.5)
so n/2(n9) > 8(1.5^n)/(1.5)
n/2(n9) > 16(1.5^n)
n^2  9n > 32  32(.5^n)
32(.5)^n + n^2  9n  32 > 0
set it equal to zero and attempt to solve it.
This would be a very nasty equation to solve, except you know that n has to be a whole number, so do some trial and error calculations.
eg. n = 2 we get 38 > 0 which is false
if n = 5 we get 44 > 0 which is false
if n = 20 we get 188 > 0 which is true
So somewhere between n=5 and n=20 there should be a solution
(n=12 > 4.0078 > 0 mmmmhhh?)