Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
As we have the first term of GP. Let r be the common ratio.
∴ we can say that GP is 1 , r , r2 , r3 … ∞
As per the condition, each term is the sum of all terms which follow it.
If a1,a2 , … represents first, second, third term etc
∴ we can say that:
a1 = a2 + a3 + a4 + …∞
⇒ 1 = r + r2 + r3 +…∞
Note: You can take any of the cases like a2 = a3 + a4 + .. all will give the same result.
We observe that the above progression possess a common ratio. So it is a geometric progression.
Common ratio = r and first term (a) = r
Sum of infinite GP = ,where a is the first term and k is the common ratio.
Note: We can only use the above formula if |k|<1
∴ we can use the formula for the sum of infinite GP.
⇒ S =
⇒
⇒ r=1−r
∴ 2r=2 or r= 1/2
Hence the series is 1, 1/2, 1/4, 1/8, 1/16...............