Find an infinite

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...............