If a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP., prove that a,

1.	If a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP., prove that a, b, c are in AP.
Answer» Here, we know a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP Also, a(1/b + 1/c) + 1, b(1/c + 1/a) + 1, c(1/a + 1/b) + 1 are in AP Let us take LCM for each expression then we get, (ac + ab + bc)/bc , (ab + bc + ac)/ac, (cb + ac + ab)/ab are in AP 1/bc, 1/ac, 1/ab are in AP Let us multiply numerator with ‘abc’, we get abc/bc, abc/ac, abc/ab are in AP ∴ a, b, c are in AP. Hence proved.

If a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP., prove that a, b, c are in AP.

Answer»

Here, we know a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP

Also, a(1/b + 1/c) + 1, b(1/c + 1/a) + 1, c(1/a + 1/b) + 1 are in AP

Let us take LCM for each expression then we get,

(ac + ab + bc)/bc , (ab + bc + ac)/ac, (cb + ac + ab)/ab are in AP

1/bc, 1/ac, 1/ab are in AP

Let us multiply numerator with ‘abc’, we get

abc/bc, abc/ac, abc/ab are in AP

∴ a, b, c are in AP.

Hence proved.