State, true or false: if a, b and c are in A.P. then :

(i) 4a, 4b and 4c are in A.P.

(ii) a + 4, b + 4 and c + 4 are in A.P.

Given,

a, b and c are in A.P.

∴ b - a = c - b

⇒ b + b = c + a

⇒ 2b = a + c ……….(1)

(i) To prove,

4a, 4b and 4c are in A.P., difference between consecutive terms should be same.

⇒ 4b - 4a = 4c - 4b

⇒ 4(b - a) = 4(c - b)

⇒ b - a = c - b

⇒ 2b = c + a, which is equal to equation 1.

Hence, proved that 4a, 4b and 4c are in A.P.

(ii) To prove,

a + 4, b + 4 and c + 4 are in A.P., difference between consecutive terms should be same.

⇒ (b + 4) - (a + 4) = (c + 4) - (b + 4)

⇒ b - a + 4 - 4 = c - b + 4 - 4

⇒ b - a = c - b

⇒ 2b = c + a, which is equal to equation 1.

Hence, proved that a + 4, b + 4 and c + 4 are in A.P.