If P(k) = k^2(k + 3) (k^2– 1) is true, then what is P(k + 1) ? - Vidyadip

MCQ

If $P(k) = k^2(k + 3) (k^2– 1)$ is true, then what is $P(k + 1)\ ?$

A
$ (k+1)^2(k+3)\left(k^2-1\right) $
B
$ (k+1)^2(k+4)\left(k^2-1\right) $
✓
$ (k+1)^2(k+4) k(k+2) $
D
$ (k+1)(k+4) k(k+2) $

✓

Answer

Correct option: C.

$ (k+1)^2(k+4) k(k+2) $

In mathematical induction, if $P(k)$ is true,
we need to prove that $P(k + 1)$ is also true.
Here $P(k + 1)$ is found by substituting $(k + 1)$ in place of $k.$
$P(k + 1) = (k + 1)^2(k + 1 + 3) ((k + 1)^2– 1)$
$P(k + 1) = (k + 1)^2(k + 4) (k^2+ 1 + 2k – 1)$
$P(k + 1) = (k + 1)^2(k + 4) (k^2+ 2k)$
$P(k + 1) = (k + 1)^2(k + 4) k (k +2)$

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