Principle of Mathematical Induction: Statement, Proof and Examples

The Principle of Mathematical Induction is a fundamental mathematical technique used to prove statements or formulas that are asserted to be true for all natural numbers. It is a powerful and elegant method that leverages the well-ordered nature of the set of natural numbers to establish the validity of an infinite sequence of propositions.

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Principle of Mathematical Induction

Mathematical induction is one of the methods which can be used to prove a variety of mathematical statements which are formulated in terms of $n$, where $n$ is a positive integer (natural number)

Suppose there is a given statement P(n) involving the natural number n such that

1. The statement is true for $n = 1$, i.e., $P(1)$ is true, and

2. If the statement is true for $n = k$ (where $k$ is some positive integer), then the statement is also true for $n = k + 1$, i.e., truth of $P(k)$ implies the truth of $P (k + 1).$

Then, $P(n)$ is true for all natural numbers $n$.

Property (i) is simply a statement of fact. There may be situations when a statement is true for all $n ≥ 2$. In this case, step 1 will start from $n = 2$ and we shall verify the result for $n = 2$, i.e., $P(2)$.

Property (ii) is a conditional property. It does not assert that the given statement is true for $n = k$, but only that if it is true for $n = k$, then should also be true for $ n = k +1$ for this principle to be applicable. So, to prove that the property holds, only prove that conditional proposition:

If the statement is true for $n = k$, then it is also true for $n = k + 1.$

Prove the statement:

$\mathrm{P}(\mathrm{n})=1+2+3+\ldots \ldots+\mathrm{n}=\frac{\mathrm{n}(\mathrm{n}+1)}{2}$ is true for all natural numbers.

Proof:

Step 1: Check if $P(1)$ is true
For $\mathrm{n}=1$, we have

$
\begin{array}{ll}
\mathrm{P}(1): & 1=\frac{1(1+1)}{2} \\
\Rightarrow & 1=1
\end{array}
$
So, formula is true for $n=1$.So $P(1)$ is true

Step 2: Assume $P(k)$ to be true and using that check if $P(k+1)$ is true
Suppose that $P(k)$ is true for some positive integer $k$

$
\mathrm{P}(\mathrm{k})=1+2+3+\ldots \ldots+\mathrm{k}=\frac{\mathrm{k}(\mathrm{k}+1)}{2}
$
Now we need to prove that $P(k+1)$ is also true.

$
\mathrm{P}(\mathrm{k}+1)=1+2+3+\ldots \ldots+(\mathrm{k}+1)=\frac{(\mathrm{k}+1)(\mathrm{k}+2)}{2}
$

we already have,

$
1+2+3+\ldots \ldots+\mathrm{k}=\frac{\mathrm{k}(\mathrm{k}+1)}{2}
$

add $(k+1)$ both side in eq (i)

$
\begin{array}{ll}
\Rightarrow & 1+2+3+\ldots \ldots+(k+1)=\frac{\mathrm{k}(\mathrm{k}+1)}{2}+(\mathrm{k}+1) \\
\Rightarrow & 1+2+3+\ldots \ldots+(\mathrm{k}+1)=\frac{(\mathrm{k}+1)}{2}(\mathrm{k}+2)
\end{array}
$

Therefore, $P(k+1)$ is true and the inductive proof is now completed.
Hence $P(n)$ is true for all natural numbers $n$.

Recommended Video Based on Mathematical Induction

Solved Examples Based on Mathematical Induction:

Example 1: What can we deduce from the following mathematical expression? $x^2=9$
1) $x=3$
2) $x=\sqrt{9}$
3) $x= \pm 3$
4) Both (B) and (C)

Solution

Principle of Mathematical Induction -

Mathematical induction is one of the techniques which can be used to prove a variety of mathematical statements which are formulated in terms of $n$, where $n$ is a positive integer

Principle of Mathematical Induction

Principle of Mathematical Induction

$
\begin{aligned}
& x^2=9 \\
& \Rightarrow x^2-9=0 \Rightarrow(x-3)(x+3)=0
\end{aligned}
$
Thus $x= \pm 3$

Example 2: To prove a result by mathematical induction what values do we need to check for n ?
1) $n=1, n=k$
2) $n=1, n=k, n=k+1$
3) $n=1, n=k-1, n=k$
4) Both (B) and (C)

Solution

Principle of Mathematical Induction

Suppose there is a given statement $P(n)$ involving the natural number n such that
1. The statement is true for $n=1$, i.e., $P(1)$ is true, and
2. If the statement is true for $\mathrm{n}=\mathrm{k}$ (where k is some positive integer), then the statement is also true for $n=k+1$, i.e., truth of $P(k)$ implies the truth of $P(k+1)$.

Then, $P(n)$ is true for all natural numbers $n$.
Property (ii) is a conditional property. It does not assert that the given statement is true for $n=k$, but only that if it is true for $n=k$, then it is also true for $\mathrm{n}=\mathrm{k}+1$. So, to prove that the property holds, only prove that conditional proposition:

If the statement is true for $\mathrm{n}=\mathrm{k}$, then it is also true for $\mathrm{n}=\mathrm{k}+1$.
As long as the terms are consecutive like ( $k, k+1$ ) or ( $k-1, k)$, Principle of Mathematical Induction can be applied

So, both B and C options are correct

Example 3: The sum $\sum_{k=1}^n(1+2+3+4+\ldots+k)$ is a polynomial in $n$ of degree
1) 2
2) 3
3) 4
4) 5

Solution

$
\sum_{k=1}^n \frac{k^2+k}{2}=\frac{1}{2}\left(\sum k^2+\sum k\right)
$

which is a polynomial of degree 2

Example 4: The expression $15^4-7^4$ is divisible by
1) 3
2) 6
3) 9
4) 11

Solution
Principle of Mathematical Induction -

$
\begin{aligned}
& 15^4-7^4=(15+7)(15-7)\left(15^2+7^2\right) \\
& =22 * 8 *(225-49) \\
& =22 * 8 * 176
\end{aligned}
$
Since $\left(15^2+7^2\right)$ is divisible by 11

Example 5: The expression $7^{2 n}+2^{3 n-3} * 3^{n-1}$ is divisible by
1) $49$
2) $64$
3) $25$
4) $36$

Solution

Principle of Mathematical Induction

Mathematical induction is one of the techniques which can be used to prove a variety of mathematical statements which are formulated in terms of $n$, where n is a positive integer

For $\mathrm{n}=1,7^2+2^0 * 3^0=49+1=50$
It is divisible by 25
Let $\mathrm{P}(\mathrm{r})$ be true for $\mathrm{n}=\mathrm{r}$

$
7^{2 r}+2^{3 r-3} \cdot 3^{r-1}=25 k
$

For $\mathrm{n}=\mathrm{r}+1$,

$
\begin{aligned}
& \Rightarrow 49.7^{2 r}+2^{3 r} \cdot 3^r=49.7^{2 r}+8.2^{3 r-3} \cdot 3.3^{r-1} \\
& =49.7^{2 r}+24\left(25 k-7^{2 r}\right)_{(\text {From (i)) }} \\
& =25\left(7^{2 r}+24 k\right)
\end{aligned}
$

Summary
Mathematical induction is one of the techniques which can be used to prove a variety of mathematical statements which are formulated in terms of $n$, where n is a positive integer. The Principle of Mathematical Induction is a cornerstone of mathematical proof techniques, enabling the validation of infinite sequences of propositions. Its elegance lies in proving a universal truth by verifying a base case and a simple inductive step. The method's versatility extends across numerous mathematical domains, proving invaluable in establishing fundamental theorems and solving complex problems.