Prove that x(x+1)(x=5) is divisible by 3 using PMI.

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Arun · Answer

let P(n)= n(n+ 1)(n + 5) P(1) = 1* 2* 6 = 12, divisible by 3 let P(k) be true P(k) = k * (k +1) *(k +5) on solving P(k) = k^3 + 6k^2 + 5k = 3 y (multiple of 3) k^3 = 3 y – 6k^2 –5k …........(1) now P(k + 1) = (k + 1) (k + 1+ 1) (k + 1+ 5) = (k + 1) (k + 2) (k + 6) on solving = k^3 + 9k^2 + 20 k +12 put value of k^3 from eqn (1) = 3y –6k^2 – 5k +9k^2 + 20k + 12 = 3y + 3k^2 +15 k +12 = 3(y + k^2 + 5k +4) therefore we can say P(k + 1) is a multiple of 3. hope it helps

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Prove that x(x+1)(x=5) is divisible by 3 using PMI.

Prove that x(x+1)(x=5) is divisible by 3 using PMI.

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