Question

Let P(n) be the statement : 2ⁿ ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ∈ N? 

Answer 1

\[P\left( n \right): 2^n \geq 3n\]
\[\text{ We know that } P\left( r \right) \text{ is true }  . \]
\[\text { Thus, we have: }  \]
\[ 2^r \geq 3r\]
\[\text{ To show: P(r + 1) is true }  . \]
\[\text{ We know: } \]
\[P(r) \text{ is true } . \]
\[ \therefore 2^r \geq 3r\]
\[ \Rightarrow 2^r . 2 \geq 3r . 2 \left[ \text{ Multiplying both sides by } 2 \right]\]
\[ \Rightarrow 2^{r + 1} \geq 6r\]
\[ \Rightarrow 2^{r + 1} \geq 3r + 3r\]
\[ = 2^{r + 1} \geq 3r + 3 \left[ \text{ Since } 3r \geq 3 \text{ for all } r \in N \right]\]
\[ = 2^{r + 1} \geq 3\left( r + 1 \right) \]
\[\text{ Hence, P(r + 1) is true } . \]
\[\text{ However, we cannot conclude that }  P\left( n \right) \text{ is true for all n } \in N . \]
\[P(1): 2^1 \not\geq 3 . 1\]
\[\text{ Therefore } , P\left( n \right) \text{ is not true for all n }  \in N .\]