Roots progression?

Algebra Level 4

If $\color{#D61F06}a , \color{#3D99F6}b , \color{#69047E}c$ are in Arithmetic Progression. Then $\dfrac{1}{\sqrt{\color{#3D99F6}b} + \sqrt{\color{#69047E}c}} , \dfrac{1}{\sqrt{\color{#69047E}c} + \sqrt{\color{#D61F06}a}} , \dfrac{1}{\sqrt{\color{#D61F06}a} + \sqrt{\color{#3D99F6}b}}$ are in ..................

Clarification : $\color{#D61F06}a , \color{#3D99F6}b , \color{#69047E}c > 0$

None of these Arithmetic - geometric progression Geometric progression Arithmetic progression Harmonic progression

1 solution

Pulkit Gupta
Nov 27, 2015

a , b & c are in an A.P. Thus, b - a = c - b implying { $\sqrt{b}$ - $\sqrt{a}$ } { $\sqrt{b}$ + $\sqrt{a}$ } = { $\sqrt{c}$ - $\sqrt{b}$ } { $\sqrt{c}$ + $\sqrt{b}$ }

We claim that the given terms are in A.P. Then, the difference of two consecutive terms should be same. On using the above equation coupled with this fact; we can easily prove that our claim is indeed true.

@Pulkit Gupta Take $a = x - y , b = x , c = x + y$

Then rationalize the given terms and you'll directly see that they are in AP.

Ankit Kumar Jain - 4 years, 2 months ago

Roots progression?

1 solution

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