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.?
- Pramod KumarLv 72 months agoFavourite answer
Here is the solution.
- 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.