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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