Use a/k + a/k(1-1/k)+.....a/k(1-1/k)^n+....=a to show that
3/4 +3/16 + 3/64 + ....+ 3/4^n....= 1
Given formula: $\displaystyle \frac{a}{k} \left[1 + \left(1 - \frac{1}{k}\right) + \, .... \, + \left(1 - \frac{1}{k}\right)^n + \, .... \right] = a$.
Required to find: $\displaystyle \frac{3}{4} \left(1 + \frac{1}{4} + \, .... \, + \frac{1}{4^{n-1}} + \frac{1}{4^n} + \, .... \right)$.
So substitute a = 1 and k = 4/3 into the given formula ......