First slide

Introduction to P.M.I

Question

For all $n \in N, n (n + 1) (n + 5)$ is a multiple of

Moderate

A

4
B

3
C

5
D

7

Solution

Let the given statement be P(n).
$i.e. P (n) : n (n + 1) (n + 5)$ is a multiple of 3.
Step l: For n=1,
$1 (1 + 1) (1 + 5) = 1 \times 2 \times 6 = 12 = 3 \times 4$
which is a multiple of 3, that is true
Step lI: Let it is true for n = k,
$k (k + 1) (k + 5) = 3 λ$
$\begin{array}{l} \Rightarrow k (k^{2} + 5 k + k + 5) = 3 λ \\ \Rightarrow k^{3} + 6 k^{2} + 5 k = 3 λ - - - - (i) \end{array}$
Step III: For n=k+1,
$\begin{array}{l} (k + 1) (k + 1 + 1) (k + 1 + 5) \\ = (k + 1) (k + 2) (k + 6) = (k^{2} + 2 k + k + 2) (k + 6) \\ = (k^{2} + 3 k + 2) (k + 6) = k^{3} + 6 k^{2} + 3 k^{2} + 18 k + 2 k + 12 \\ = (k^{3}) + 9 k^{2} + 20 k + 12 \\ = (3 λ - 6 k^{2} - 5 k) + 9 k^{2} + 20 k + 12 [using Eq. (i)] \\ = 3 λ + 3 k^{2} + 15 k + 12 = 3 (λ + k^{2} + 5 k + 4) \end{array}$
which is a multiple of 3.
Therefore, P(k + 1) is true when P(k) is true.
Hence, from the principle of mathematical induction, the statement is true for all natural numbers n.

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