If a, b, c are in A.P., show that : (b + c), (c + a) and (a + b) are also in A.P.

Given,

a, b and c are in A.P.

∴ b - a = c - b

⇒ b + b = a + c

⇒ 2b = a + c

If (b + c), (c + a) and (a + b) are in A.P., then

⇒ (a + b) - (c + a) = (c + a) - (b + c)

⇒ a + b - c - a = c + a - b - c

⇒ b - c = a - b

⇒ b + b = a + c

⇒ 2b = a + c

Since, 2b = a + c is proved above.

Hence, proved that (b + c), (c + a) and (a + b) are also in A.P.