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RRB NTPC Mains or Stage 2 Model Paper 5 with solutions

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Question : 15 of 120

Marks: +1, -0

What is the value of

\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\dots+\frac{1}{\sqrt{15}+\sqrt{16}}

?

0
1
2
3

Solution:

Concept:

Use the conjugate to simplify each reciprocal surd term.

Explanation:

For any positive integer

n

,

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

So the sum becomes:

(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+\cdots+(\sqrt{16}-\sqrt{15})

All intermediate terms cancel.

Only

\sqrt{16}-1

remains.

Since

\sqrt{16}=4

,

\sqrt{16}-1=4-1=3

Answer:

3

, so the correct option is D.

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