Question

using the principle mathematical induction prove that
n(n+1)(n+5) is multiple of 3​

Answer 1

$\to$ P(n) = n(n+1)(n+5) is a multiple of 3

For P(1)

$\to$ P(1) = 1 ( 1 + 1 ) ( 1 + 5 ) = 2 × 6 = 12

{ True , 12 Is A multiple of 3 }

Let , this is also true for P(k)

$\to \: P(k) = k ( k + 1 ) ( k + 5 ) = 3m \\ \\ \to \: P(k) = ( k + 1 ) = \frac{3m}{k(k + 5)}$

{ m = Integer }

Now For P( K + 1)

$\to \: P(k+1) = (k+1)(k+2)(k+6) \\ \\ \to \: </p><p>p(k+1) = \frac{(3m )\times (k + 2)(k + 6)}{k(k + 5)} \\ \\ \to \: p(k+1) = 3 \: ( \frac{m(k + 2)(k + 6)}{k(k + 5)} )$

Hence , P(k+1) is true whenever P(k) is true

Thus , P(k+1) is true whenever P(k) is true Thus , from PMI , P(n) is true for all natural number

Answer 2

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