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  4. The remainder on dividing 1+3+32+33+ ldots+32021 by 50 is .

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Q. The remainder on dividing $1+3+3^{2}+3^{3}+\ldots+3^{2021}$ by $50$ is _____.

JEE MainJEE Main 2022Binomial Theorem

Solution:

$\frac{1 .\left(3^{2022}-1\right)}{2}=\frac{9^{1011}-1}{2}$
$=\frac{(10-1)^{1011}-1}{2}$
$=\frac{100 \lambda+10110-1-1}{2}$
$=50 \lambda+\frac{10108}{2}$
$=50 \lambda+5054$
$=50 \lambda+50 \times 101+4$
Rem $(50)=4 .$