3 Marks Question

Question 13 Marks

If $P(n)$ is the statement " $n^2+n$ is even", and if $P(r)$ is true, then $P(r+1)$ is true.

Answer

$P(n): n^2+n$ is even
Given, P(r) is true
$\Rightarrow r^2 + r $is even
$\Rightarrow \text{r}^2 + \text{r} = 2\lambda \ ...(1)$
Now,
$(r + 1)^2 + (r + 1)$
$= r^2 + 1 + 2r + r + 1$
$= (r + 1)^2 + 2r + 2$
$=2\lambda + 2\text{r} + 2$ [Using equation (1)]
$=2(\lambda + \text{r} + 1)$
$=2\lambda$
$\Rightarrow (r + 1)^2 + (r + 1)$ is even
$\Rightarrow P(r + 1)$ is true

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Question 23 Marks

If P(n) is the statement "$n^2 - n + 41$ is prime", prove that $P(1), P(2)$ and $P(3)$ are true. Prove also that $P(41)$ is not true.

Answer

$P(n): n^2-n+41$ is prime
$P(1): 1-1+41$ is prime
$\Rightarrow P(1): 41$ is prime
$\therefore P (1)$ is true.
$P (2): 2^2-2+41$ is prime
$\Rightarrow P(2): 43$ is prime
$\therefore P (2)$ is true.
$P(3): 3^2-3+41$ is prime
$\Rightarrow P(3): 47$ is prime
$\therefore P(3)$ is true.
$P (41)$ : $(41)^2-41+41$ is prime
$P(41)$ : $(41)^2$ is prime
$\Rightarrow P(41)$ is not true.

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Question 33 Marks

If P(n) is the statement $"2^n \geq 3n"$ and if $P(r)$ is true, prove that $P(r + 1)$ is true.

Answer

$P(n): 2^n \geq 3n$
Given that P(r) is true
$\Rightarrow 2^r \geq 3r$
Multiplying both sides by $2,$
$2.2^r \geq 2.3r$
$2^{r+1} \geq 6r$
$2^{r+1} \geq 3r + 3r$
$2^{r+1} \geq 3 + 3r$, [Since $3r \geq 3 \Rightarrow 3r + 3r \geq 3 + 3r]$
$2^{r+1} \geq 3r(r + 1)$
$\Rightarrow P(r + 1)$ is true.

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