CAT 2017 Slot 2 — QA Question 33

The given series is an infinite series.
Let first term a₁ = a and the common ratio be r.

Now, a_n = 3(a_n+1 + a_n+2 + ……..)
⇒ ar^n-1 = 3(arⁿ + arⁿ⁺¹ + ...)
⇒ ar^n-1 = 3$\left(\frac{{\mathrm{ar}}^{\mathrm{n}}}{1-\mathrm{r}}\right)$
⇒ 1 - r = 3r
⇒ r = 1/4

Now, 32 = a₁ + a₂ + a₃ + ...
⇒ 32 = a + ar + ar² + ar³ + ...
⇒ 32 = $\frac{\mathrm{a}}{1-\mathrm{r}}$ = $\frac{\mathrm{a}}{1-1/4}$ = $\frac{4\mathrm{a}}{3}$
⇒ a = 24

∴ a₅ = ar⁴ = $24\times {\left(\frac{1}{4}\right)}^4$ =  $\frac{24}{256}$ = $\frac{3}{32}$

Hence, option (c).