# Sum of this geometric expression

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#1
I need to find an expression for the sum of this series in terms of r, N and a.
N
∑ar^2n (a and r are both constants)
n=0

I assume that I need to use the equation for the sum of a geometric series and I think I'll have to add +a to the fraction as the series starts at n=0 rather than n=1, but I'm struggling because the power is not in the standard form. If it was just n then I'd use n+1, but as it's multiplied by 2 I assume I can't just stick 2n+1 into the equation? I can't figure out how to deal with the 2n.

I'm probably missing something really obvious here but any help would be appreciated.
Last edited by Tammie2345524; 3 weeks ago
0
3 weeks ago
#2
(Original post by Tammie2345524)
I need to find an expression for the sum of this series in terms of r, N and a.
N
∑ar^2n (a and r are both constants)
n=0

I assume that I need to use the equation for the sum of a geometric series and I think I'll have to add +a to the fraction as the series starts at n=0 rather than n=1, but I'm struggling because the power is not in the standard form. If it was just n then I'd use n+1, but as it's multiplied by 2 I assume I can't just stick 2n+1 into the equation? I can't figure out how to deal with the 2n.

I'm probably missing something really obvious here but any help would be appreciated.
You can think of this series as being a + a(r^2) + a(r^2)^2 + a(r^2)^3 etc.
1
3 weeks ago
#3
As above, but another trick to help reduce confusion about starting / end indices is to recognise that the geometric sum formula can be written as
[ first term - (last+1)th term] / (1 - r)
So here the first term is "a" the (last+1)th term is "ar^2(N+1)" and the ratio of successve terms, the usual r, is "r^2".

This is similar to the more familiar arithemetic sum (gauss) which is
(first term + last term)*n/2
Last edited by mqb2766; 3 weeks ago
1
#4
(Original post by old_engineer)
You can think of this series as being a + a(r^2) + a(r^2)^2 + a(r^2)^3 etc.
(Original post by mqb2766)
As above, but another trick is help reduce confusion about starting / end indices is to recognise that the geometric sum formula can be written as
[ first term - (last+1)th term] / (1 - r)
So here the first term is "a" the (last+1)th term is "ar^2(N+1)" and the ratio of successve terms, the usual r, is "r^2".

This is similar to the more familiar arithemetic sum (gauss) which is
(first term + last term)*n/2
Ohhh, thanks guys I think I've got it now 0
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