1. ## Induction help

I understand the rules of prooving by induction, but I can't understand the math in this equation.
If someone could explain what is happening at each step I would be very thankful.

$\displaystyle k + 1 <= 3^k +1 <= 3^k + 3^k = 2.3^k <= 3.3^k = 3^k^+^1$

The way I do math is down the page not across
1)$\displaystyle k + 1 <= 3^k +1$
is the equation $\displaystyle 3^k + 3^k$ a simplification of the RHS of the above equation (1)?
how do we get $\displaystyle 3^k + 3^k$ from $\displaystyle 3^k +1$ ??

thanks for any help.

2. Hello dunsta
Originally Posted by dunsta
I understand the rules of prooving by induction, but I can't understand the math in this equation.
If someone could explain what is happening at each step I would be very thankful.

$\displaystyle k + 1 <= 3^k +1 <= 3^k + 3^k = 2.3^k <= 3.3^k = 3^k^+^1$

The way I do math is down the page not across
1)$\displaystyle k + 1 <= 3^k +1$
is the equation $\displaystyle 3^k + 3^k$ a simplification of the RHS of the above equation (1)?
how do we get $\displaystyle 3^k + 3^k$ from $\displaystyle 3^k +1$ ??

thanks for any help.
If $\displaystyle k\ge0$, then:
$\displaystyle 3^k \ge 1$

$\displaystyle \Rightarrow 1 \le 3^k$
So if we add $\displaystyle 3^k$ to both sides of this inequality:
$\displaystyle 3^k+1\le3^k+3^k$
But
$\displaystyle 3^k+3^k=2.3^k$
So
$\displaystyle 3^k+1\le2.3^k$
And obviously
$\displaystyle 2<3$
So
$\displaystyle 3^k+1\le3.3^k$
And
$\displaystyle 3.3^k=3^{k+1}$
So
$\displaystyle 3^k+1\le3^{k+1}$
OK now?