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Function Operation Summation

XAT 2010 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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The operation (x)(x) is defined by (i) (1)=2(1) = 2 (ii) (x+y)=(x)⋅(y)(x + y) = (x) \cdot (y) for all positive integers x and y. If ∑x=1n(x)=1022\sum_{x=1}^{n} (x) = 1022 then n=n =

  1. A.

    8

  2. B.

    9

  3. C.

    10

  4. D.

    11

  5. E.

    None of the above

Answer

B

Explanation

From the functional equation, f(x)=2xf(x) = 2^x. ∑x=1n2x=2+22+23+...+2n=2(2n−1)2−1=2n+1−2\sum_{x=1}^{n} 2^x = 2 + 2^2 + 2^3 + ... + 2^n = \frac{2(2^n - 1)}{2 - 1} = 2^{n+1} - 2 Given 2n+1−2=1022  ⟹  2n+1=1024=210  ⟹  n+1=10  ⟹  n=92^{n+1} - 2 = 1022 \implies 2^{n+1} = 1024 = 2^{10} \implies n + 1 = 10 \implies n = 9.

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