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Question

The weighted arithmetic mean ofthe first n natural numbers whose weights are equal to the corresponding numbers is given by,

A
12(n+1)
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B
12(n+2)
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C
13(2n+1)
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D
13n(2n+1)
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Solution

The correct option is B 13(2n+1)
xi x Weight (w)
x1 1 1
x2 2 2
x3
....
....
3 3
xn n n
First n natural numbers are written in the table along with their respective weight.
Weighted arithmetic mean =r1nwrxrr1nwr
=1×1+2×2+3×3+...+n×n1+2+3+4+...+n

=12+22+32+...+n21+2+3+4+...+n

Using formula of sum of series, we get
=n(n+1)(2n+1)6×2n(n+1)

=(2n+1)3
Hence, this is the required answer.

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