Question

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

 

Answer 1

We have: \[P(n): 2^n \geq 3n\]
\[\text{ Also } , \]
\[P(r) \text{ is true } . \]
\[ \therefore 2^r \geq 3r\]
\[\text{ To Prove } : P(r + 1)\text{  is true } . \]
\[\text{ We have } : \]
\[ 2^r \geq 3r\]
\[ \Rightarrow 2^r \times 2 \geq 3r \times 2 \left[ \text{ Multiplying both sides by 2 }  \right]\]
\[ \Rightarrow 2^{r + 1} \geq 6r\]
\[ \therefore 2^{r + 1} \geq 3r + 3 \left[ 6r \geq 3r + 3 \text{ for every } r \in N . \right]\]
\[\text{ Hence, P(r + 1) is true }  .\]