Find the weighted mean of the first n natural | vedclass

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(B) The weighted mean is given by $\bar{x}_w = \frac{\sum_{i=1}^{n} i \cdot i^2}{\sum_{i=1}^{n} i^2}$.This simplifies to $\bar{x}_w = \frac{\sum_{i=1}

Vedclass · Accepted Answer

$\frac{3n(n + 1)}{2(2n + 1)}$

Vedclass · Answer

(B) The weighted mean is given by $\bar{x}_w = \frac{\sum_{i=1}^{n} i \cdot i^2}{\sum_{i=1}^{n} i^2}$.This simplifies to $\bar{x}_w = \frac{\sum_{i=1}^{n} i^3}{\sum_{i=1}^{n} i^2}$.Using the standard summation formulas $\sum i^3 = [\frac{n(n+1)}{2}]^2$ and $\sum i^2 = \frac{n(n+1)(2n+1)}{6}$,we get:$\bar{x}_w = \frac{[n(n+1)/2]^2}{n(n+1)(2n+1)/6} = \frac{n^2(n+1)^2}{4} \cdot \frac{6}{n(n+1)(2n+1)}$.Simplifying the expression,we obtain $\bar{x}_w = \frac{3n(n+1)}{2(2n+1)}$.

Marks	$2$	$3$	$5$	$7$
Frequency	$(x+1)^2$	$2x-5$	$x^2-3x$	$x$

Marks obtained	No. of students
$0-10$	$2$
$10-20$	$18$
$20-30$	$30$
$30-40$	$45$
$40-50$	$35$
$50-60$	$20$
$60-70$	$6$
$70-80$	$3$

Find the weighted mean of the first $n$ natural numbers,where the weights are equal to their squares.

Similar Questions

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If for some positive $x \in R$,the frequency distribution of the marks obtained by $20$ students in a certain test is as follows:
Marks $2$ $3$ $5$ $7$
Frequency $(x+1)^2$ $2x-5$ $x^2-3x$ $x$

Then the mean of the marks is:

What is the median of an ordered series of $n$ even observations?

Compute the median from the following table:

Marks obtained No. of students

$0-10$ $2$

$10-20$ $18$

$20-30$ $30$

$30-40$ $45$

$40-50$ $35$

$50-60$ $20$

$60-70$ $6$

$70-80$ $3$

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