Prove that 11-2-1-2-3-1-3-4- - -1-8-9-2

We have, 11+√(2)+1√(2)+√(3)+1√(3)+√(4)+ ............. +1√(8)+√(9) =11+√(2)×1 √(2)1 √(2)+1√(2)+√(3)×√(2) √(3)√(2) √(3)+1√(3)+√(4)×√(3) √(4)√(3) √(4)+ .. ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Summation by Sigma Method

Prove that ...

Question

Prove that $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + . . . . . . . . . . . . . + \frac{1}{\sqrt{8} + \sqrt{9}}$ =2

Open in App

Solution

We have,
$\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + . . . . . . . . . . . . . + \frac{1}{\sqrt{8} + \sqrt{9}}$

$= \frac{1}{1 + \sqrt{2}} \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} \times \frac{\sqrt{3} - \sqrt{4}}{\sqrt{3} - \sqrt{4}} + . . . . . . . . . . . . . + \frac{1}{\sqrt{8} + \sqrt{9}} \times \frac{\sqrt{8} - \sqrt{9}}{\sqrt{8} - \sqrt{9}}$

$= \frac{1 - \sqrt{2}}{- 1} + \frac{\sqrt{2} - \sqrt{3}}{- 1} + . . . . . . . . . . . . . . . . . . . . . . . + \frac{\sqrt{8} - \sqrt{9}}{- 1}$

$= \sqrt{2}$ $- 1$ $+ \sqrt{3}$ $- \sqrt{2}$ $+ \sqrt{4}$ $- \sqrt{3}$ $+ \sqrt{5}$ $- \sqrt{4}$ $+ \sqrt{6}$ $- \sqrt{5}$ $+ \sqrt{7} - \sqrt{6} + \sqrt{8} - \sqrt{7} + \sqrt{9} - \sqrt{8}$

$= - 1 + 3 = 2$

Suggest Corrections

Similar questions

Q. Prove that

\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{5}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}}

= 2

Prove that $\frac{9 π}{8} - \frac{9}{4} s i n^{- 1} (\frac{1}{3}) = \frac{9}{4} s i n^{- 1} (\frac{2 \sqrt{2}}{3})$

Q. Prove that

\frac{9 π}{8} - \frac{9}{4} {sin}^{- 1} (\frac{1}{3}) = \frac{9}{4} {sin}^{- 1} (\frac{2 \sqrt{2}}{3})

Q. Prove:

\frac{9 π}{8} - \frac{9}{4} {sin}^{- 1} \frac{1}{3} = \frac{9}{4} {sin}^{- 1} \frac{2 \sqrt{2}}{3}

Q. Show that

\frac{9 π}{8} - \frac{9}{4} {sin}^{- 1} (\frac{1}{3}) = \frac{9}{4} {sin}^{- 1} (\frac{2 \sqrt{2}}{3})

Join BYJU'S Learning Program

Prove that 11+√2+1√2+√3+1√3+√4+.............+1√8+√9=2

Prove that $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + . . . . . . . . . . . . . + \frac{1}{\sqrt{8} + \sqrt{9}}$ =2