Show that if the positive number a, b, c are in A.P. Show that the numbers 1/​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​(√b + √c),

Question

Show that if the positive number a, b, c are in A.P. Show that the numbers

\(\frac{1}{\sqrt{b}+\sqrt{c}}\), \(\frac{1}{\sqrt{c}+\sqrt{a}}\), \(\frac{1}{\sqrt{a}+\sqrt{b}}\) will be in A.P.

Answer 1

\(\frac{1}{\sqrt{b}+\sqrt{c}} \), \(\frac{1}{\sqrt{c}+\sqrt{a}} \), \(\frac{1}{\sqrt{a}+\sqrt{b}} \) will be in A. P.

If \(\frac{1}{\sqrt{c}+\sqrt{a}} \) − \(\frac{1}{\sqrt{b}+\sqrt{c}} \) = \(\frac{1}{\sqrt{a}+\sqrt{b}} \) − \(\frac{1}{\sqrt{c}+\sqrt{a}} \)

i.e if \(\frac{\sqrt{b}-\sqrt{a}}{(\sqrt{c}+\sqrt{a})(\sqrt{b}+\sqrt{c})} \) = \(\frac{\sqrt{c}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})(\sqrt{c}+\sqrt{a})} \)

i.e if \(\frac{\sqrt{b}-\sqrt{a}}{(\sqrt{b}+\sqrt{c})} \) = \(\frac{\sqrt{c}-\sqrt{b}}{\sqrt{a}+\sqrt{b}} \)

i.e if b − a = c + b

i.e if 2b = a + c

i.e if a, b, c are in A.P.

Thus, a, b, c are in A.P. ⇒ \(\frac{1}{\sqrt{b}+\sqrt{c}} \), \(\frac{1}{\sqrt{c}+\sqrt{a}} \), \(\frac{1}{\sqrt{a}+\sqrt{b}} \) are in A. P.