The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then find its first term.

Given, a_n = 128, r = 2 and S_n = 255.

By formula a_n = ar^{n - 1}
∴ 128 = a(2)^{n - 1}

$\Rightarrow a\dfrac{2^n}{2} = 128 \\[1em] \Rightarrow 2^n = \dfrac{256}{a}. \text{ (Eq 1) }$

By formula,

$S_n = \dfrac{a(r^n - 1)}{r - 1} \\[1em] \therefore 255 = \dfrac{a(2^n - 1)}{2 - 1}$

Putting value of 2ⁿ from Eq 1 in above Equation,

$\Rightarrow 255 = a\Big(\dfrac{256}{a} - 1\Big) \\[1em] \Rightarrow 255 = 256 - a \\[1em] \Rightarrow a = 256 - 255 = 1.$

Hence, the first term of the G.P. is 1.