Question
Class 11MathematicsSequences and Series

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

Verified Answer

We have:

a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in A.P.

⇒ (a/b + a/c), (b/c + b/a), (c/a + c/b) are in A.P.

Adding 1 to each term:

(a/b + a/c + 1), (b/c + b/a + 1), (c/a + c/b + 1) are in A.P.

⇒ (ac + ab + bc)/bc, (ab + bc + ca)/ca, (bc + ca + ab)/ab are in A.P.

Multiplying each term by (abc)/(ab + bc + ca), we get a, b, c are in A.P.

Hence proved: a, b, c are in A.P.