જો $A=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]$ અને $M=A+A^{2}+A^{3}+\ldots .+A^{20}$ આપેલ હોય તો શ્રેણિક $\mathrm{M}$ ના બધાજ ઘટકોનો સરવાળો મેળવો.
JEE MAIN 2021, Difficult
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$A^{n}=\left[\begin{array}{lll}1 & n & \frac{n^{2}+n}{2} \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right]$
So, required sum
$=20 \times 3+2 \times\left(\frac{20 \times 21}{2}\right)+\sum_{\mathrm{r}=1}^{20}\left(\frac{\mathrm{r}^{2}+\mathrm{r}}{2}\right)$
$=60+420+105+35 \times 41=2020$
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