Solve the Following Question.(3 Marks)

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

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

Answer

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

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Maths Solve the Following Question.(3 Marks)

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