Question 2: Reduce the expression \(\mathrm{M}=1+3+3^{2}++\ldots+3^{118}+3^{119}\)
\(\mathrm{M}=1+3+3^{2}++\ldots+3^{118}+3^{119}\)
\(\begin{aligned} &3 \mathrm{M}=3 .\left(1+3+3^{2}++\ldots . .+3^{118}+3^{119}\right)= 3+3^{2}+\ldots .+3^{119}+3^{120} \\ &3 \mathrm{M}-\mathrm{M}=\left(3+3^{2}+\ ldots . .+3^{119}+3^{120}\right)-\left(1+3+3^{2}++\ldots+3^{118}+3^{119}\right) \\ &2 \mathrm{M}=3^{120}-1 \\ &M=\frac{3^{120}-1}{2} \end{aligned}\)
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