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If P (n) is the statement

Question:

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

Solution:

We have:

$P(n): 2^{n} \geq 3 n$

Also,

$P(r)$ is true.

 

$\therefore 2^{r} \geq 3 r$

To Prove: $P(r+1)$ is true.

 

We have:

 $2^{r} \geq 3 r$

$\Rightarrow 2^{r} \times 2 \geq 3 r \times 2 \quad$ [Multiplying both sides by 2]

 

$\Rightarrow 2^{r+1} \geq 6 r$

$\therefore 2^{r+1} \geq 3 r+3 \quad[6 r \geq 3 r+3$ for every $r \in N .]$

Hence, $P(r+1)$ is true.

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