3^rd term of a G.P. is 27 and its 6^th term is 729; find the product of its first and 7^th terms.

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

We know that,

n^th term of G.P. = ar^{n - 1}

Given,

3^rd term of a G.P. = 27

∴ ar² = 27 ……….(1)

Given,

6^th term of a G.P. = 729

∴ ar⁵ = 729 ……….(2)

Dividing equation (2) by (1), we get :

$\Rightarrow \dfrac{ar^5}{ar^2} = \dfrac{729}{27} \\[1em] \Rightarrow r^3 = 27 \\[1em] \Rightarrow r^3 = 3^3 \\[1em] \Rightarrow r = 3.$

Substituting value of r in equation (1), we get :

⇒ a(3)² = 27

⇒ 9a = 27

⇒ a = $\dfrac{27}{9}$

⇒ a = 3.

7th term = ar⁶

= 3(3)⁶

= 3 × 729

= 2187.

Product of first and seventh term = 3 x 2187

= 6561.

Hence, product of first and seventh term = 6561.