G.P. : $\dfrac{2}{9}, \dfrac{1}{3}, \dfrac{1}{2},…………..$

Assertion (A): 5^th of the given G.P. is $1\dfrac{1}{8}$.

Reason (R): If for a G.P., the first term is a, the common ratio is r and the number of terms = n, then sum of the first n term S_n = $\dfrac{a(r^n - 1)}{r - 1}$ for all r.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

Given, the sequence = $\dfrac{2}{9}, \dfrac{1}{3}, \dfrac{1}{2},…………..$

First term (a) = $\dfrac{2}{9}$

Common ratio (r) = $\dfrac{\dfrac{1}{3}}{\dfrac{2}{9}} = \dfrac{1 \times 9}{2 \times 3} = \dfrac{3}{2}$

Using the formula; T_n = a.r^{n - 1}

$T_5 = \dfrac{2}{9} × \Big(\dfrac{3}{2}\Big)^{5 - 1}\\[1em] = \dfrac{2}{9} × \Big(\dfrac{3}{2}\Big)^4\\[1em] = \dfrac{2}{9} × \dfrac{81}{16}\\[1em] = \dfrac{9}{8}\\[1em] = 1\dfrac{1}{8}.$

So, assertion (A) is true.

When first term = a, common ratio = r then

Sum of first n terms (S_n) = $\dfrac{a(r^n - 1)}{r - 1}$

So, reason (R) is true, but it doesnot explain assertion.

Hence, option 4 is the correct option.