JEE Main 2026 Jan 28 Shift 2, Mathematics Q20: Arithmetic, Geometric And Harmonic Progressions

Separate the first term and treat the rest as a GP. The tail $\frac{10\cdot 1}{3^{25}}+\frac{10\cdot 2}{3^{24}}+\cdots +\frac{10\cdot 2^{24}}{3}$ has first term $\frac{10}{3^{25}}$, common ratio $r=\frac{2/3^{-1}}{1/3^0}=6$ (each successive term: numerator multiplied by 2, denominator divided by 3 — net factor 6), and 25 terms.

Sum of tail $=\frac{10}{3^{25}}\cdot \frac{6^{25}-1}{6-1}=\frac{10}{3^{25}}\cdot \frac{6^{25}-1}{5}=\frac{2(6^{25}-1)}{3^{25}}=2\cdot \frac{6^{25}}{3^{25}}-\frac{2}{3^{25}}=2\cdot 2^{25}-\frac{2}{3^{25}}=2^{26}-\frac{2}{3^{25}}$.

Total $S=\frac{6}{3^{26}}+2^{26}-\frac{2}{3^{25}}=\frac{2}{3^{25}}+2^{26}-\frac{2}{3^{25}}=2^{26}$.