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If a,b,c in A.P. prove that. a×a(b+c),b×b(c+a),c×c(a+b) are in A.P.
Given, a,b and c are in A.P.a2(b + c), b2(c + a), c2(a + b) will be in A.P.if b2(c + a) – a2(b + c) = c2(a + b) – b2(c + a)i.e., if c(b2 – a2) + ab(b – a) = a(c2 – b2) + bc(c– b)i.e., if (b – a)(ab + bc + ca) = (c – b)(ab + bc + ca)i.e., if b – a = c – bi.e., if 2b = a + ci.e., if a, b, c are in A.P.Thus, if a, b, c are in A.P. then a2(b + c), b2(c + a), c2(a + b) will also be in A.P.
Given, a,b and c are in A.P.
a2(b + c), b2(c + a), c2(a + b) will be in A.P.
if b2(c + a) – a2(b + c) = c2(a + b) – b2(c + a)
i.e., if c(b2 – a2) + ab(b – a) = a(c2 – b2) + bc(c– b)
i.e., if (b – a)(ab + bc + ca) = (c – b)(ab + bc + ca)
i.e., if b – a = c – b
i.e., if 2b = a + c
i.e., if a, b, c are in A.P.
Thus, if a, b, c are in A.P. then a2(b + c), b2(c + a), c2(a + b) will also be in A.P.
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