If the first term of a G.P. is 729 and its 7th term is 64, then the su

1.	If the first term of a G.P. is 729 and its 7th term is 64, then the sum of the first seven terms is(a) 2187 (b) 2059 (c) 1458 (d) 2123
Answer» (b) 2059 Let the first term and common ratio of the G.P. be a and r respectively. Then, T₁= a = 729, T₇ = ar⁶ = 64 ∴ \(\frac{ar^6}{a}\) = \(\frac{64}{729}\) ⇒ r⁶= \(\frac{64}{729}\) = \(\frac{2^6}{3^6}\) ⇒ r = \(\frac{2}{3}.\) ∴ S₇ = \(\frac{a(1-r^n)}{1-r}\) = \(\frac{729\bigg(1-\big(\frac{2}{3}\big)^7\bigg)}{1-\frac{2}{3}}\) = \(\frac{729\bigg(1-\frac{128}{2187}\bigg)}{\frac{1}{3}}\) = 2187 \(\bigg(\frac{2187-128}{2187}\bigg)\) = 2059.

If the first term of a G.P. is 729 and its 7th term is 64, then the sum of the first seven terms is(a) 2187 (b) 2059 (c) 1458 (d) 2123

Answer»

(b) 2059

Let the first term and common ratio of the G.P. be a and r respectively.

Then, T₁= a = 729, T₇ = ar⁶ = 64

∴ \(\frac{ar^6}{a}\) = \(\frac{64}{729}\) ⇒ r⁶= \(\frac{64}{729}\) = \(\frac{2^6}{3^6}\) ⇒ r = \(\frac{2}{3}.\)

∴ S₇ = \(\frac{a(1-r^n)}{1-r}\) = \(\frac{729\bigg(1-\big(\frac{2}{3}\big)^7\bigg)}{1-\frac{2}{3}}\)

= \(\frac{729\bigg(1-\frac{128}{2187}\bigg)}{\frac{1}{3}}\) = 2187 \(\bigg(\frac{2187-128}{2187}\bigg)\) = 2059.