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NDA 1 2024 Maths, question 117

Topic: Statistics

Passage · Q117 to Q118

xix_{i} 11 22 33 …\ldots nn
fif_{i} 11 2−12^{-1} 2−22^{-2} …\ldots 2−(n−1)2^{-(n-1)}
Q117

What is ∑inxifi\sum _{i}^{n}x_{i}f_{i} equal to ?

  1. (a) 2n+1−n+22n−1\frac{2^{n+1}-n+2}{2^{n-1}}
  2. (b) 2n+1−n−22n−1\frac{2^{n+1}-n-2}{2^{n-1}}Correct answer
  3. (c) 2n+1+n+22n−1\frac{2^{n+1}+n+2}{2^{n-1}}
  4. (d) 2n+1−n−22n\frac{2^{n+1}-n-2}{2^{n}}

Solution.

Answer: (b) 2n+1−n−22n−1\frac{2^{n+1}-n-2}{2^{n-1}}

  1. Let S = 1 + 2/2 + 3/2² + ··· + n/2^(n−1).
  2. Subtract S/2 from S: S/2 = 1 + 1/2 + ··· + 1/2^(n−1) − n/2ⁿ = 2 − (n + 2)/2ⁿ.
  3. Thus S = [2^(n+1) − n − 2]/2^(n−1).