Question

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

Answer 1

Answer: 4

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