Question

#ello bello every one

Plz help meh ​

Answer 1

Answer:

$\frac{1}{\sqrt{a_{1} }+\sqrt{a_{2} } } +\frac{1}{\sqrt{a_{2} }+\sqrt{a_{3} } }+...\frac{1}{\sqrt{a_{n-1} }+\sqrt{a_{n} } } = \frac{n-1}{\sqrt{a_{1} }+\sqrt{a_{n} } }\\\\L.H.S\\\\\frac{1}{\sqrt{a_{1} }+\sqrt{a_{2} } } +\frac{1}{\sqrt{a_{2} }+\sqrt{a_{3} } }+...\frac{1}{\sqrt{a_{n-1} }+\sqrt{a_{n} } }$

Rationalize denominator

$\frac{\sqrt{a_{2}}-\sqrt{a_{1}} }{a_{2}-a_{1} } + \frac{\sqrt{a_{3}}-\sqrt{a_{2}}}{a_{3}-a_{2} } +... +\frac{\sqrt{a_{n}}-\sqrt{a_{n-1}}}{a_{n}-a_{n-1} }\\\\\frac{\sqrt{a_{2}}-\sqrt{a_{1}} }{d } + \frac{\sqrt{a_{3}}-\sqrt{a_{2}}}{d } +... +\frac{\sqrt{a_{n}}-\sqrt{a_{n-1}}}{d }\\\\\\\frac{\sqrt{a_{n}}-\sqrt{a_{1}}}{d}$

multiply and divide by  $\\\sqrt{a_{n}}+\sqrt{a_{1}}$

$\\\frac{(\sqrt{a_{n}}-\sqrt{a_{1}})(\sqrt{a_{n}}+\sqrt{a_{1}})}{d(\sqrt{a_{n}}+\sqrt{a_{1}})}\\\frac{a_{n}-a_{1}}{d(\sqrt{a_{n}}+\sqrt{a_{1}})}\\\\\frac{[ a_{1}+(n-1)d ]-a_{1}}{d(\sqrt{a_{n}}+\sqrt{a_{1}})}\\\\\frac{(n-1)d}{d(\sqrt{a_{n}}+\sqrt{a_{1}})}\\\\\frac{(n-1)}{(\sqrt{a_{n}}+\sqrt{a_{1}})}$

L.H.S = R.H.S

Hence, proved.

MARK ME AS RAINLIEST(it took lot of time to type. despite that I gave you an answer)!! GOOD LUCK :))