# i struggle with proofs by induction

• December 12th 2009, 05:39 PM
i struggle with proofs by induction
Prove by induction that for all integers n greater than or equal to 2, 2^n+1 < 3^n
• December 12th 2009, 06:36 PM
craig
Quote:

Prove by induction that for all integers n greater than or equal to 2, 2^n+1 < 3^n

This is how I would do it:

For $n=2$, $2^3 < 3^2$, $8<9$, true for $n=2$

Assume for $n=k$, this gives you $2^{k+1} < 3^k$

Now the induction step, prove for $n=k+1$

$2^{k+2} < 3^{k+1}$

$2 \cdot 2^{k+1} < 3 \cdot 3^{k}$

Dividing both sides by 2

$2^{k+1} < \frac{3}{2} \cdot 3^{k}$

Therefore if $2^{k+1} < 3^k$, then $2^{k+1}$ is definitely $< \frac{3}{2} \cdot 3^{k}$, true for all $n=k+1$ if true for $n=k$

True for $n=2$, therefore true for all integers greater than or equal to 2.

QED
• December 15th 2009, 04:07 PM