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?

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Solution

$P (n) : 2^{n} \geq 3 n We k n o w t h a t P (r) is true . Th u s, w e h a v e : 2^{r} \geq 3 r T o s h o w : P (r + 1) is true . W e k n o w : P (r) is true . ∴ 2^{r} \geq 3 r \Rightarrow 2^{r} . 2 \geq 3 r . 2 [Multiplying both sides by 2] \Rightarrow 2^{r + 1} \geq 6 r \Rightarrow 2^{r + 1} \geq 3 r + 3 r = 2^{r + 1} \geq 3 r + 3 [Since 3 r \geq 3 for all r \in N] = 2^{r + 1} \geq 3 (r + 1) Hence, P (r + 1) is true . However, we cannot conclude that P (n) is true for all n \in N . P (1) : 2^{1} ≱ 3.1 Therefore, P (n) is not true for all n \in N .$

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