JEE Main 2026 — 24 January, Evening Shift — Question 7

Let $\mathrm{a}=\frac{4}{7}, \mathrm{~b}=\frac{1}{3}$

Multiply $\mathrm{N}^{\mathrm{r}}$ and $\mathrm{D}^{\mathrm{r}}$ by $(\mathrm{a}-\mathrm{b})=\frac{4}{7}-\frac{1}{3}=\frac{5}{21}$

$\frac{1}{a-b}\left[\left(a^{2}-b^{2}\right)+\left(a^{3}-b^{3}\right)+\left(a^{4}-b^{4}\right)+\ldots \infty\right]$

$\frac{1}{\mathrm{a}-\mathrm{b}}\left[\frac{\mathrm{a}^{2}}{1-\mathrm{a}}-\frac{\mathrm{b}^{2}}{1-\mathrm{b}}\right]=\frac{21}{5}\left[\frac{\frac{16}{49}}{1-\frac{4}{7}}-\frac{\frac{1}{9}}{1-\frac{1}{3}}\right]$

$=\frac{21}{5}\left[\frac{16}{21}-\frac{1}{6}\right]=\frac{21}{5}\left[\frac{96-21}{21.6}\right]$

$=\frac{75}{5.6}=\frac{15}{6}=\frac{5}{2}$