Q10 of 176 Page 8

If a, b, c are in A.P., prove that:

a3 + c3 + 6abc = 8b3

Given: a, b, c are in AP


∴ a + c = 2b …(i)


…(ii)


Taking Lhs i.e. a3 + c3 + 6abc


[from (i)]


⇒ a3 + c3 + 3ac(a + c)


⇒ a3 + c3 3a2c + 3ac2


⇒ (a + c)3


⇒ (2b)3 [from (ii)]


= 8b3 = RHS


Hence Proved


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If (b – c)2, (c – a)2, (a – b)2 are in A.P., then show that: are in A.P.

[Hint: Add ab + bc + ca — a2 — b2 — c2 to each term or let = b — c, = c— a, = a — b, then + + = 0]

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If a, b, c are in A.P., prove that:

(a — c)2 = 4 (a — b)(b — c)

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If a, b, c are in A.P., prove that:

(a + 2b — c)(2b + c — a)(c + a — b) = 4abc


[Hint: Put b = on L.H.S. and R.H.S.]

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