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JEE Main 202225 JuneEvening Shift
Exam
JEE Main
Year
2022
Date
25 June
Shift
Evening Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2022 · 25 June · Evening Shift · Mathematics · Sequences and Series

The sum 1 + 2 · 3 + 3 · 3 ^ 2 + . . + 10 · 3 ^ 9 is equal to

The sum 1+2⋅3+3⋅32+……..+10⋅391 + 2 \cdot 3 + 3 \cdot {3}^{2} + \ldots \ldots . . + 10 \cdot {3}^{9} is equal to

Options

  1. A2⋅312+104\dfrac{2 \cdot {3}^{12} + 10}{4}
  2. B19⋅310+14\dfrac{19 \cdot {3}^{10} + 1}{4}
  3. C5⋅310−25 \cdot {3}^{10} - 2
  4. D9⋅310+12\dfrac{9 \cdot {3}^{10} + 1}{2}

Correct answer

B. 19⋅310+14\dfrac{19 \cdot {3}^{10} + 1}{4}

Step-by-step solution

Let S=1⋅30+2⋅31+3⋅32+…..+10.39S = 1 \cdot {3}^{0} + 2 \cdot {3}^{1} + 3 \cdot {3}^{2} + \ldots . . + 10 . {3}^{9} ....(i)

3S=1.31+2.32……………….+9×39+10×3103S = 1 . {3}^{1} + 2 . {3}^{2} \ldots \ldots \ldots \ldots \ldots \ldots . + 9 \times {3}^{9} + 10 \times {3}^{10} .....(ii)

Now subtracting equation (ii) from (i) we get,

−2S=(1⋅30+31+32…..39)−10.310- 2S = \left(1 \cdot {3}^{0} + {3}^{1} + {3}^{2} \ldots . . {3}^{9}\right) - 10 . {3}^{10}

S=5×310−(310−14)S = 5 \times {3}^{10} - \left(\dfrac{{3}^{10} - 1}{4}\right)

S=20.310−310+14=19.310+14S = \dfrac{20 . {3}^{10} - {3}^{10} + 1}{4} = \dfrac{19 . {3}^{10} + 1}{4}

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