NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

The value of the sum $\frac{1}{2\sqrt{1} + 1\sqrt{2}} + \frac{1}{3\sqrt{2} + 2\sqrt{3}} + \frac{1}{4\sqrt{3} + 3\sqrt{4}} + \dots + \frac{1}{25\sqrt{24} + 24\sqrt{25}}$ is:

Choose the correct answer:

Explanation

1. Identify the General Term

Looking at the pattern of the sum, we can write the $n^{th}$ term ( $T_n$ ) as:

T_n = \frac{1}{(n+1)\sqrt{n} + n\sqrt{n+1}}

where $n$ goes from $1$ to $24$ .

2. Rationalize the General Term

To simplify $T_n$ , we multiply the numerator and the denominator by the conjugate $(n+1)\sqrt{n} - n\sqrt{n+1}$ :

T_n = \frac{(n+1)\sqrt{n} - n\sqrt{n+1}}{((n+1)\sqrt{n} + n\sqrt{n+1})((n+1)\sqrt{n} - n\sqrt{n+1})}

The denominator is in the form $(a+b)(a-b) = a^2 - b^2$ :

Denominator = (n+1)^2(n) - n^2(n+1)

Denominator = n(n+1) \left[ (n+1) - n \right]

Denominator = n(n+1)(1) = n(n+1)

So, the simplified general term is:

T_n = \frac{(n+1)\sqrt{n} - n\sqrt{n+1}}{n(n+1)}

3. Split the Term

Now, divide each part of the numerator by the denominator:

T_n = \frac{(n+1)\sqrt{n}}{n(n+1)} - \frac{n\sqrt{n+1}}{n(n+1)}

T_n = \frac{\sqrt{n}}{n} - \frac{\sqrt{n+1}}{n+1}

T_n = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}

4. Calculate the Total Sum

This is a telescoping series. Let's sum $T_n$ from $n=1$ to $n=24$ :

S = \sum_{n=1}^{24} \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \right)

Expanding the terms:

S = \left( \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}} \right) + \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}} \right) + \dots + \left( \frac{1}{\sqrt{24}} - \frac{1}{\sqrt{25}} \right)

All intermediate terms cancel out, leaving only the first and the last term:

S = \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{25}}

S = 1 - \frac{1}{5}

S = \frac{4}{5}

Correct Option: (b)