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

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