Question

# The positive integer $$n$$ for which $$2. { 2 }^{ 2 }+3. { 2 }^{ 3 }+4. { 2 }^{ 4 }+\cdots +n. { 2 }^{ n }={ 2 }^{ n+10 }$$ is

A
510
B
511
C
512
D
513

Solution

## The correct option is B $$513$$$$2^{n+10} = 2\cdot2^2 + 3\cdot2^3+4\cdot2^4+\cdots+n\cdot2^n\rightarrow(1)$$Multiply $$(1)$$ by $$2$$$$2^{n+11} = 2\cdot2^3 + 3\cdot2^4+4\cdot2^5+\cdots+n\cdot2^{n+1}\rightarrow(2)$$Subtracting $$(2)$$ from $$(1)$$ by shifting one place, we get$$-2^{n+10} = 2\cdot2^2 + 2^3+2^4+\cdots+2^n - n\cdot2^{n+1}\rightarrow(3)$$$$n\cdot2^{n+1} - 2^{n+10} = 2 + 2^1 + 2^2 + \cdots+2^n$$$$\Rightarrow n\cdot2^n - 2^9\cdot2^n = 1 + \dfrac{2^n-1}{2-1}$$$$\Rightarrow n-1=2^9$$$$n = 513$$Maths

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