# Mathematical Induction Inequality

Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for subtraction and/or greatness, using the assumption in step 2. Let’s take a look at the following hand-picked examples.

## Basic Mathematical Induction Inequality

Prove $4^{n-1} \gt n^2$ for $n \ge 3$ by mathematical induction.

Step 1:  Show it is true for $n=3$.
LHS $=4^{3-1} = 16$
RHS $=3^2=9$
LHS > RHS
Therefore it is true for $n=3$.
Step 2:  Assume that it is true for $n=k$.
That is, $4^{k-1} > k^2$.
Step 3:  Show it is true for $n=k+1$.
That is, $4^{k} > (k+1)^2$.
\begin{aligned} \displaystyle \require{ASMSymbols} \require{color} \text{LHS } &= 4^k \\ &= 4^{k-1+1} \\ &= 4^{k-1} \times 4 \\ &\gt k^2 \times 4 &\color{red}{ \text{by the assumption } 4^{k-1} > k^2} \\ &= k^2 + 2k^2 + k^2 &\color{red}{2k^2 > 2k \text{ and } k^2 > 1 \text{ for } k \ge 3} \\ &\gt k^2 + 2k + 1 \\ &= (k+1)^2 \\ &=\text{RHS} \\ \text{LHS } &\gt \text{ RHS} \end{aligned}
Therefore it is true for $n=k+1$, assuming that it is true for $n=k$.
Therefore $4^{n-1} \gt n^2$ is true for $n \ge 3$.

## Mathematical Induction Inequality using Differences

Prove $n^2 \lt 2^n$ for $n \ge 5$ by mathematical induction.

It is quite often used to prove $A > B$ by $A-B >0$.

Step 1:  Show it is true for $n=5$.
LHS $= 5^2 = 25$
RHS $= 2^5 = 32$
LHS $\lt$ RHS
It is true for $n=5$.
Step 2:  Assume that it is true for $n=k$.
That is, $k^2 \lt 2^k$.
Step 3:  Show it is true for $n=k+1$.
That is, $(k+1)^2 \lt 2^{k+1}.$
\begin{aligned} \displaystyle \require{ASMSymbols} \require{color} \text{RHS } – \text{ LHS } &= 2^{k+1}-(k+1)^2 \\ &= 2 \times 2^k-(k^2+2k+1) \\ &\gt 2 \times k^2-(k^2+2k+1) &\color{red}{\text{ by the assumption from Step 2}} \\ &= k^2-2k-1 \\ &= (k-1)^2-2 \\ &\gt 0 &\color{red}{\text{since } k \ge 5 \text{ and so } (k-1)^2 \ge 16} \\ 2^{k+1}-(k+1)^2 &\gt 0 \\ (k+1)^2 &\lt 2^{k+1} \\ \end{aligned}
Therefore it is true for $n=k+1$, assuming it is true for $n=k$.
Therefore it is true for $n=k+1$ is true for $n \ge 5$. 1. GEORGE ANTASHYAN

Thank you. I would like to see more problems like these if possible.

1. iitutor Post author

Hi George!
Thank you for reaching us. We are so glad that our mathematics contents helped you. You may find more interesting mathematical inductions here: https://iitutor.com/free-mathematics-lessons/. New contents are being added weekly.
Thanks again.

2. Venetta Scurti

Thank you for your exquisite explanations.