Question

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

Answer 1

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 = \(\frac{a}{1-k}\) ,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.

Hence the series is 1, 1/2, 1/4, 1/8, 1/16.........