GATE CS 2013 Q.61 (GA Q.61): Quantitative Aptitude

Find the sum of the expression

\dfrac{1}{\sqrt{1} + \sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + \dfrac{1}{\sqrt{3} + \sqrt{4}} + \dots + \dfrac{1}{\sqrt{80} + \sqrt{81}}

Official answer and solution

Solution

Rationalize the denominator of each term to obtain a telescoping sum.

Consider the general term of the given series:

T_k = \dfrac{1}{\sqrt{k} + \sqrt{k+1}}

Rationalizing the denominator by multiplying both numerator and denominator by $(\sqrt{k+1} - \sqrt{k})$ :

T_k = \dfrac{\sqrt{k+1} - \sqrt{k}}{(\sqrt{k+1} + \sqrt{k})(\sqrt{k+1} - \sqrt{k})} = \dfrac{\sqrt{k+1} - \sqrt{k}}{(k+1) - k} = \sqrt{k+1} - \sqrt{k}

Now sum this expression from $k = 1$ to $k = 80$ :

S = \sum_{k=1}^{80} (\sqrt{k+1} - \sqrt{k})

S = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + \dots + (\sqrt{81} - \sqrt{80})

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

S = \sqrt{81} - \sqrt{1} = 9 - 1 = 8

Thus, the sum is 8, which corresponds to option (B).

Final answer

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