a, b and c are in A.P. Prove that b + c, c + a and a + b are in A.P.

∵ a, b and c are in A.P.

∴ a + c = 2b

If, (a + b) + (b + c) + (c + a) are in AP then, (a + b) + (b + c) = 2(a + c)

∴ To prove- (a + b) + (b + c) = 2(a + c)

Now,

(a + b) + (b + c)

= a + 2b + c

= a + a + c + c

[ ∵ a + c = 2b]

= 2a + 2c

=2(a + c)

⇒ (a + b) + (b + c) + (c + a) are in AP

Hence, Proved