Skip to content
GATETECH
Official PYQ · 2013Q61MCQ1 mark

GATE CS 2013 Q61 — General Aptitude

Explain with AI

Find the sum of the expression

11+2+12+3+13+4+............+180+81\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+............+\frac{1}{\sqrt{80}+\sqrt{81}}

Choose one option
View worked solution

Answer: B

To find the sum of the given expression, we can rationalize the denominator of the general term.

Let the kk-th term of the series be:

Tk=1k+k+1T_k = \frac{1}{\sqrt{k} + \sqrt{k+1}}

Multiplying the numerator and the denominator by the conjugate (k+1−k)(\sqrt{k+1} - \sqrt{k}):

Tk=k+1−k(k+1+k)(k+1−k)=k+1−k(k+1)−k=k+1−kT_k = \frac{\sqrt{k+1} - \sqrt{k}}{(\sqrt{k+1} + \sqrt{k})(\sqrt{k+1} - \sqrt{k})} = \frac{\sqrt{k+1} - \sqrt{k}}{(k+1) - k} = \sqrt{k+1} - \sqrt{k}

Now, we can write the sum SS of the series from k=1k = 1 to 8080 as:

S=T1+T2+T3+⋯+T80S = T_1 + T_2 + T_3 + \dots + T_{80} S=(2−1)+(3−2)+(4−3)+⋯+(81−80)S = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + \dots + (\sqrt{81} - \sqrt{80})

This is a telescoping series where all the intermediate terms cancel out:

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

Therefore, the sum of the expression is 8.

Free practice · Your answers on this page are not saved.

Published
Report a correction
Keep exploringMore General Aptitude questions