proving that $\frac{1}{\sqrt a + \sqrt b} +\frac{1}{\sqrt b + \sqrt c} + \frac{1}{\sqrt c + \sqrt d} = \frac{3}{\sqrt a + \sqrt d} $ for any A.P.

Question

if $a,b,c,d$ are an arithmetic progression (in that order), prove that $$\frac{1}{\sqrt a + \sqrt b} +\frac{1}{\sqrt b + \sqrt c} + \frac{1}{\sqrt c + \sqrt d} = \frac{3}{\sqrt a + \sqrt d} $$

I made $n$ the common difference of $a,b,c,d$; so $$a=a$$ $$b=a +n$$ $$c=a + 2n$$ $$d=a+3n$$ I tried to replace the terms with those, anyways i squared both equalities but i didn 't get nothing since i'm pretty bad with square roots. I'm looking for some hints or properties that can be useful. Thanks

Answer 1

HINT

Multiply and divide by conjugate of each denominator, then you'll get a $(-n)$ in each denominator.

Then : $$\sqrt{a}-\sqrt{b}~+\sqrt{b}-\sqrt{c}~+\sqrt{c}-\sqrt{d} \over {-n}$$

$$=\frac{\sqrt{a} -\sqrt{d}}{-n}=\frac{a-d}{-n \cdot (\sqrt{a} + \sqrt{d})}= \frac{3}{\sqrt{a} + \sqrt{d}}$$

Answer 2

$$\frac{1}{\sqrt a + \sqrt b} +\frac{1}{\sqrt b + \sqrt c} + \frac{1}{\sqrt c + \sqrt d} =\frac{1}{n}(\sqrt{b}-\sqrt{a}+\sqrt{c}-\sqrt{b}+\sqrt{d}-\sqrt{c})=$$ $$=\frac{d-a}{n(\sqrt{a}+\sqrt{d})}= \frac{3}{\sqrt a + \sqrt d} $$

Answer 3

$\frac{1}{\sqrt a + \sqrt b} +\frac{1}{\sqrt b + \sqrt c} + \frac{1}{\sqrt c + \sqrt d} = \frac{3}{\sqrt a + \sqrt d} $

Make into A.P.

$S =\frac{1}{\sqrt a + \sqrt {a+x}} +\frac{1}{\sqrt {a+x} + \sqrt {a+2x}} + \frac{1}{\sqrt {a+2x} + \sqrt {a+3x}} - \frac{3}{\sqrt a + \sqrt {a+3x}} $

Rationalize

$\frac{1}{\sqrt a + \sqrt {a+x}}\frac{\sqrt a - \sqrt {a+x}}{\sqrt a - \sqrt {a+x}} =\frac{\sqrt a - \sqrt {a+x}}{-x} $

$\frac{1}{\sqrt {a+x} + \sqrt {a+2x}}\frac{\sqrt {a+x} - \sqrt {a+2x}}{\sqrt {a+x} - \sqrt {a+2x}} =\frac{\sqrt {a+x} - \sqrt {a+2x}}{-x} $

$\frac{1}{\sqrt {a+2x} + \sqrt {a+3x}}\frac{\sqrt {a+2x} - \sqrt {a+3x}}{\sqrt {a+2x} - \sqrt {a+3x}} =\frac{\sqrt {a+2x} - \sqrt {a+3x}}{-x} $

$\frac{3}{\sqrt a + \sqrt {a+3x}}\frac{\sqrt a - \sqrt {a+3x}}{\sqrt a - \sqrt {a+3x}} =\frac{3\sqrt a -3 \sqrt {a+3x}}{-3x} =\frac{\sqrt a - \sqrt {a+3x}}{-x} $

Combine

$\begin{array}\\ S &= (-1/x) ((\sqrt a - \sqrt {a+x}) +(\sqrt {a+x} - \sqrt {a+2x})\\ &\qquad +(\sqrt {a+2x} - \sqrt {a+3x}) -(\sqrt a - \sqrt {a+3x}))\\ &= 0 \qquad\text{because everything cancels out!!!}\\ \end{array} $