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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
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B
511
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C
512
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D
513
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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
$$-2^{n+10} = 2\cdot2^2 + 2^3+2^4+\cdots+2^n - n\cdot2^{n+1}\rightarrow(3)$$

$$n\cdot2^{n+1} - 2^{n+10} = 2 + 2^1 + 2^2 + \cdots+2^n$$

$$\Rightarrow n\cdot2^n - 2^9\cdot2^n = 1 + \dfrac{2^n-1}{2-1}$$

$$\Rightarrow n-1=2^9$$

$$n = 513$$

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