The positive integer n for which $$2$$×$$22+3$$×$$23+4$$×$$24+$$…$$+n$$×$$2n=2n+10$$ is

Question

The positive integer n for which $$2$$×$$22+3$$×$$23+4$$×$$24+$$…$$+n$$×$$2n=2n+10$$ is

Infinity Learn · Accepted Answer

We have                    $$2n+10=2$$×$$22+3$$×$$23+4$$×$$24+$$…$$+n$$×$$2n$$⇒        $$22n+10=2$$×$$23+3$$×$$24+$$…$$+(n$$−$$1)$$×$$2n+$$ n×$$2n+1$$                         Subtracting, we get             −$$2n+10=2$$⋅$$22+23+24+$$…$$+2n$$−n⋅$$2n+1$$                       $$=8+82n$$−$$2$$−$$12$$−$$1$$−n×$$2n+1$$                       $$=2n+1$$−$$(n)2n+1$$⇒            $$210=2n$$−$$2$$ ⇒ $$n=513$$

The positive integer n for which $2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} = 2^{n + 10}$ is

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The positive integer n for which 2×22+3×23+4×24+…+n×2n=2n+10 is

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The positive integer n for which $2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} = 2^{n + 10}$ is