The sum 1+23+33^2++103^9 is equal to:

Question

The sum $1+2\cdot3+3\cdot3^2+\cdots+10\cdot3^9$ is equal to:

Accepted Answer

Let $S = 1\cdot3^0+2\cdot3^1+\cdots+10\cdot3^9$. Then $3S = 1\cdot3^1+2\cdot3^2+\cdots+10\cdot3^{10}$.

$$-2S = (3^0+3^1+\cdots+3^9) - 10\cdot3^{10} = \frac{3^{10}-1}{2} - 10\cdot3^{10}.$$

$$S = \frac{10\cdot3^{10}-\frac{3^{10}-1}{2}}{2} = \frac{19\cdot3^{10}+1}{4}.$$

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