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

Question

The sum $1+2 \cdot 3+3 \cdot 3^{2}+\ldots . .+10 \cdot 3^{9}$ is equal to

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Answer

b
$S =1 \cdot 3^{0}+2 \cdot 3^{1}+3 \cdot 3^{2}+\ldots . .+10.3^{9}$

$3 S =1 \cdot 3^{1}+2.3^{2} \ldots \ldots \ldots \ldots \ldots \ldots+9 \times 3^{9}+10 \times 3^{10}$

$-2 S =\left(1 \cdot 3^{0}+3^{1}+3^{2} \ldots 3^{9}\right)-10.3^{10}$

$S =5 \times 3^{10}-\left(\frac{3^{10}-1}{4}\right)$

$S =\frac{20.3^{10}-3^{10}+1}{4}=\frac{19.3^{10}+1}{4}$

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