# Enjoy Maths: The first term of an infinite GP is 1 and any term is equal to the sum of all the succeeding terms. Find the series.?

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- IndicaLv 72 months ago
This means that sum of GP = 1+1 = 2 (first term + following terms)

∴ 2=1/(1-r) ⟹ r=½

- nbsaleLv 62 months ago
1 + 1/2 + 1/4 + 1/8 + ... is pretty obvious.Let's prove it's the only series.

Let r be the common ratio, and let S be the sum.

The series after the first term just looks like r times the series, so using what's given you have:

1 = rS and therefore

S = 1/r

You always have S = first term times 1/(1-r) so you also have

S = 1/(1-r)

Therefore 1/(1-r) = 1/r

Multiply by r(1-r) and you get

r = 1-r

2r = 1

r = 1/2.

So 1 + 1/2 + 1/4 + ... is the only solution.

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