Find the sum to infinity of a decreasing G.P. with the common ratio x such that ∣ x ∣

Question

Find the sum to infinity of a decreasing G.P. with the common ratio x such that $∣ x ∣ < 1; x \neq = 0$ . The ratio of the fourth term to the second term is $\frac{1}{16}$ and the ratio of third term to the square of the second term is $\frac{1}{9}$ .

TestHub · Accepted Answer

Let the series be a , a x , a x 2 , a x 3 … given that ∣ x ∣ < 1 and x  = 0 . Also, T 2 ​ T 4 ​ ​ = a x a x 3 ​ = 16 1 ​ ⇒ x 2 = 16 1 ​ . ⇒ x = ± 4 1 ​ But since it is a decreasing G.P., ⇒ x = 4 1 ​ . Also, T 2 2 ​ T 3 ​ ​ = ( a x ) 2 a x 2 ​ = 9 1 ​ ⇒ a 1 ​ = 9 1 ​ ⇒ a = 9 S ∞ ​ = 1 − r a ​ = 1 − 4 1 ​ 9 ​ = 3 9 × 4 ​ = 12

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