First slide

Introduction to P.M.I

Question

$\frac{\frac{1}{2} \cdot \frac{2}{2}}{1^{3}} + \frac{\frac{2}{2} \cdot \frac{3}{2}}{1^{3} + 2^{3}} + \frac{\frac{3}{2} \cdot \frac{4}{2}}{1^{3} + 2^{3} + 3^{3}} + \dots . .$ up to n terms is equal to

Moderate

A

$\frac{n - 1}{n}$
B

$\frac{n}{n + 1}$
C

$\frac{n + 1}{n + 2}$
D

$\frac{n + 1}{n}$

Solution

For n =1, we have
$\frac{n}{n + 1} = \frac{1}{2} and \frac{\frac{1}{2} \cdot \frac{2}{2}}{1^{3}} = \frac{1}{2}$
For n = 2, we have
$\frac{n}{n + 1} = \frac{2}{3} and \frac{\frac{1}{2} \cdot \frac{2}{2}}{1^{3}} + \frac{\frac{2}{2} \cdot \frac{3}{2}}{1^{3} + 2^{3}} = \frac{1}{2} + \frac{3}{2 \times 9} = \frac{2}{3}$
Hence, option (b) is correct.

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