Q42 / 75 · JEE Main 2026 (28 January Shift 2) · Mathematics
$\frac{6}{3^{26}}+\frac{10 \cdot 1}{3^{25}}+\frac{10 \cdot 2}{3^{24}}+\frac{10 \cdot 2^{2}}{3^{23}}+\ldots+\frac{10 \cdot 2^{24}}{3}$ is equal to :
- Correct answerA$2^{26}$
- B$3^{25}$
- C$3^{26}$
- D$2^{25}$
Solution
The sum is $\frac{6}{3^{26}} + \sum_{r=0}^{24} \frac{10 \cdot 2^r}{3^{25-r}}$
$= \frac{6}{3^{26}} + 10 \cdot \frac{1}{3^{25}} \sum_{r=0}^{24} (2 \cdot 3)^r = \frac{6}{3^{26}} + \frac{10}{3^{25}} \cdot \frac{6^{25}-1}{5}$
$= \frac{6 + 2 \cdot 3(6^{25}-1)}{3^{26}} = \frac{6 + 2 \cdot 6^{25} \cdot 3 - 6}{3^{26}} = \frac{6 \cdot 6^{25}}{3^{26}}$
$= \frac{6^{26}}{3^{26}} = \left(\frac{6}{3}\right)^{26} = 2^{26}$
$= \frac{6}{3^{26}} + 10 \cdot \frac{1}{3^{25}} \sum_{r=0}^{24} (2 \cdot 3)^r = \frac{6}{3^{26}} + \frac{10}{3^{25}} \cdot \frac{6^{25}-1}{5}$
$= \frac{6 + 2 \cdot 3(6^{25}-1)}{3^{26}} = \frac{6 + 2 \cdot 6^{25} \cdot 3 - 6}{3^{26}} = \frac{6 \cdot 6^{25}}{3^{26}}$
$= \frac{6^{26}}{3^{26}} = \left(\frac{6}{3}\right)^{26} = 2^{26}$
Hence, the correct answer is option: 'A'.