Question

If a,b,c are in ap prove that
a^3+c^3+6abc=8b^3

Answer 1

just using the property of AP and then by cubing both sides the proof is done

Answer 2

a, b , c are in Arithmetic progression.

b - a = c - b

⇒b + b = c + a

⇒2b = (c + a). ....... (1)

⇒(2b)³ = (c + a)³

[ we know, (x + y)³ = x³ + y³ + 3x²y + 3xy² ]

⇒2b × 2b × 2b = c³ + a³ + 3c²a + 3ca²

⇒8b³ = c³ + a³ + 3ac(c + a)

[ from equation (1), ]

⇒8b³ = c³ + a³ + 3ac(2b)

⇒8b³ = c³ + a³ + 6abc [ hence proved]

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