Question

$P\left(n\right):2^{n+2}<3^n$, is true for

• A. $n\in N$
• B. $n>3,n\in N$
• C. $n>2,\forall n\in N$
• D. none of these.

Answer 1

Answer: (B)

$n>3,n\in N$

$P\left(n\right):2^{n+2}<3^n$

Let $n=1P\left(1\right):2^3<3^1$ i.e. $P\left(1\right)=8<3$ false

Let $n=2P\left(2\right):2^4<3^2$ i.e. $P\left(2\right)=16<9$ false

Let $n=3P\left(3\right):2^5<3^3$ or $P\left(3\right)=32<27,$ false

Let $n=4$

$\therefore P\left(4\right):2^6<3^4$ or $P\left(4\right)=64<81$ which is true.

$P\left(n\right):2^{n+2}<3^n$ is true for $\forall n>3,nϵN$