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JEE Main 202628 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
28 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2026 · 28 January · Evening Shift · Mathematics · Sequences and Series

6 3^ 26 + 10 · 1 3^ 25 + 10 · 2 3^ 24 + 10 · 2^ 2 3^ 23 + + 10 · 2^...

6326+10⋅1325+10⋅2324+10⋅22323+…+10⋅2243\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 :

Options

  1. A2262^{26}
  2. B3253^{25}
  3. C3263^{26}
  4. D2252^{25}

Correct answer

A. 2262^{26}

Step-by-step solution

The sum is 6326+∑r=02410⋅2r325−r\frac{6}{3^{26}} + \sum_{r=0}^{24} \frac{10 \cdot 2^r}{3^{25-r}}
=6326+10⋅1325∑r=024(2⋅3)r=6326+10325⋅625−15= \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}
=6+2⋅3(625−1)326=6+2⋅625⋅3−6326=6⋅625326= \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}}
=626326=(63)26=226= \frac{6^{26}}{3^{26}} = \left(\frac{6}{3}\right)^{26} = 2^{26}
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