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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. |
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Answer» \(\frac{1}{\sqrt{b}+\sqrt{c}} \), \(\frac{1}{\sqrt{c}+\sqrt{a}} \), \(\frac{1}{\sqrt{a}+\sqrt{b}} \) 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. |
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