Question

BEST ANS BRAINLIEST PLUS IF ANS IS WITH BEST EXPLAINATION I WILL FOLLOW THE ANSWERER...





If a, b, c are in A.P., then show that:
a^2(b + c), b^2 (c + a), c^2 (a + b) are also in A.P.​

Answer 1

Since a, b, c are in AP, we can write b as 'a + d' and c as 'a + 2d'

So, a² * (b+c) = a² * ( a + d + a+ 2d) = a²*( 2a+3d ) = 2a³+3a²d, Let this be the first term, call it A.

b²*( c+a) = (a + d)²*(a+ 2d + a) = (a²+2ad+d²)(2a+2d) = 2a³+2a²d+4a²d+4ad²+2ad²+2d³ = (2a³+3a²d) + (3a²d+ 6ad²+2d³) let it be the second term

B = A + D

c²*(a+b) = (a+2d)²*(a+a+d) = (a²+4ad+4d²)(2a+d) = 2a³+a²d+8a²d+4ad²+8ad²+4d³ = (2a³+3a²d) + (6a²d+12ad²+4d³)

= (2a³+3a²d) + 2(3a²d+6ad²+2d³) let this be the third term

C = A + 2D

By checking the values of A, B and C, we can find that they are in A.P. with first term A =2a³+3a²d and common difference D= 3a²d+ 6ad²+2d³

Hope it helps, please marks as the Brainliest answer, Cheers