<p><Success style='float: center;margin-top:16rem;text-align:center'><B>Mathematical induction lecture-1</B></Success></p>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>A proof technique</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Mathematical Induction is a proof teachnique.</p> <p>Examples:</p> <p><strong>1.</strong> Sum of first ’n’ natural numbers is $\frac{n(n+1)}{2}$</p> <p>$1+2+…..+n = \frac{n(n+1)}{2}$</p> <ol start="2"> <li>For a positive integer n,</li> </ol> <p>$n^3 -n$ is divisible by 3</p> <ol start="3"> <li>A set with n elements has $2^n$ subsets</li> </ol> </div> </main> <hr> <footer style='font-size:18px'> <span style="color:blue">A proof technique</span> $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Principle of mathematical induction</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>For given $P(n)$</p> <p><strong>Goal:</strong> $P(n)$ is true for all $ n \in \mathbb{N}$</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ <span style="color:blue">Principle of mathematical induction</span> $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Principle of mathematical induction</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>We can complete this goal in two steps:</p> <ol> <li> <p>We verify that P(1) is true</p> </li> <li> <p>Assume that P(k) is true for an arbitrary positive integer k. Prove that $p(k+1)$ is true</p> </li> </ol> <p><strong>Remark</strong></p> <p>Warning: We are not assuming that P(k) is true all k, we will assume for P(k) and then show for P(k+1).</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ <span style="color:blue">Principle of mathematical induction</span> $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Remark</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li> <p>First step - Basis step</p> </li> <li> <p>Second step - Inductive step</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ <span style="color:blue">Remark</span> $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Induction hypothesis</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Assumption that P(k) is true is generally refer to as Induction hypothesis.</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ <span style="color:blue">Induction hypothesis</span> $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Question</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Why MI is a valid proof technique?</p> <p>Not inductive reasoning</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ <span style="color:blue">Question</span> $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Show that sum of first n natural numbers is $\frac{n(n+1)}{2}$</p> <p>$i.e. 1+2+ … +n = \frac{n(n+1)}{2}$</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ <span style="color:blue">Example</span> $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Use M.I to prove</p> <p>$1+2+…+n = \frac{n(n+1)}{2}$ $\forall n \in \mathbb{N}$</p> <p>$P(n) : 1+2+…+n = \frac{n(n+1)}{2}$</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li>To prove P(n) is true for all n</li> </ul> <p><strong>Basic step:</strong></p> <ul> <li> <p>L.H.S = 1</p> </li> <li> <p>= $\frac{1(1+1)}{2}$</p> </li> <li> <p>i.e $p(1)$ is true</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Inductive step</p> <ul> <li> <p>Assume $P(k)$ is true</p> </li> <li> <p>1+2+….+k = $\frac{k(k+1)}{2}$</p> </li> <li> <p>To prove that $P(k+1)$ is true</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>To prove</p> <ul> <li>$1+2+…+k+k+1$</li> </ul> <p>$= ~ \frac{(k+1)((k+1)+1)}{2}$</p> <p>$(1+2+3+…+k)+ k+1$</p> <p>$= ~ \frac{k(k+1)}{2}+ k+1 $ (By Induction Hypothesis)</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>$\Rightarrow$ $\frac{k(k+1) + 2(k+1)}{2}$</p> <p>$\Rightarrow$ $\frac{k^2 + {3k} +2}{2}$</p> <p>$\Rightarrow$ $\frac{(k+2)(k+1)}{2}$</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li> <p>By M.I, P(n) is true for all n</p> </li> <li> <p>$\therefore$ $1+2+…+n = \frac{n(n+1)}{2} \quad \forall n\in \mathbb{N}$</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Remark</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li>MI is not a tool to get theorems about set of natural numbers.But it is a proof technique which can be used if we already conjectured the result.</li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ <span style="color:blue">Remark </span> $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Conjucture a formula for sum of first n odd positive integers. Verify your conjucture using M.I</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ <span style="color:blue">Example </span>$\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li> <p>$1+3 = 4 = 2^2$</p> </li> <li> <p>$1+3+5 = 9 = 3^2$</p> </li> <li> <p>$1+3+5+7 = 16 = 4^2$</p> </li> </ul> <p>.</p> <p>.</p> <p>.</p> <ul> <li>$1+3+5+…..+ 2n-1 = n^2$ $\leftarrow$ Conjucture</li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li> <p>To P.T.</p> </li> <li> <p>$1+3+…+2n-1 = n^2 \quad\quad \forall n \in \mathbb{N}$</p> </li> <li> <p>$P(n): 1+3+…+ 2n-1 = n^2$</p> </li> <li> <p>We wish $P(n)$ is true for all n</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Basis step</p> <ul> <li> <p>To verify $p(1)$</p> </li> <li> <p>$1 = 1^2$</p> </li> <li> <p>P(1) is true</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <p>Inductive step:</p> <ul> <li> <p>Assume P(k) is true</p> </li> <li> <p>$1+3+5+…+ 2k-1 $ = $k^2$</p> </li> <li> <p>We have to P.T. P(k+1) is true</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li>ie, we have to P.T</li> </ul> <p>$\Rightarrow$ $1+3+5+…+ 2(k)+1 = (k+1)^2$</p> <p>$\Rightarrow$ $(1+3+5+… 2k-1) + 2k+1$</p> <p>$\Rightarrow$ $( k^2 )+ 2k+1$ ( induction hypothesis)</p> <p>$\Rightarrow$ $(k+1)^2$</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$<span style="color:blue"> Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Inductive step</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li> <p>Hence by MI</p> </li> <li> <p>P(n) is true for all n</p> </li> </ul> <p>i.e</p> <ul> <li>$1+3+5+…+ 2n-1 = n^2 \quad \forall \ n \in \mathbb{N}$</li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ <span style="color:blue">Inductive step</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Remark</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left;'> <ul> <li> <p>Often, we may require to prove P(n) is true for $n = b, b+1, b+2 …..$</p> </li> <li> <p>b is some integer other than 1</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ <span style="color:blue">Remark</span> $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Remark</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <p><strong>Basic step</strong></p> <ul> <li>We verify P(b) is true</li> </ul> <p><strong>Inductive step</strong></p> <ul> <li> <p>We assume P(k)</p> </li> <li> <p>$k \in \{b, b+1 …\}$</p> </li> <li> <p>Then we prove P(k+1)</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ <span style="color:blue">Remark</span> $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <p>P.T. $1+2+…+2^n= 2^{n+1} -1 \quad \forall$ non negative integer n.</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ <span style="color:blue">Example</span> $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <p>Basic step</p> <ul> <li> <p>verify P(0) is true</p> </li> <li> <p>p(n): $1+2+…+2^n = 2^{n+1} - 1$</p> </li> <li> <p>L.H.S. $2^0 = 1$</p> </li> <li> <p>R.H.S. $2^{0+1} -1 = 2-1 =1$</p> </li> <li> <p>$\therefore$ P(0) is true</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Inductive step</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <p>Inductive step</p> <ul> <li> <p>Assume P(k) is true for k, a non negative integer</p> </li> <li> <p>To Show that P(k+1) is true</p> </li> </ul> <p>i.e</p> <ul> <li> <p>We have to Show that</p> </li> <li> <p>$1+2+…+2^{k+1} = 2^{(k+1)+1} -1= 2^{k+2} - 1$</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ Solution $\to$ <span style="color:blue">Inductive step</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <p>$(1+2+…+2^k) + 2^{k+1}$</p> <p>$\Rightarrow$ $2^{k+1} -1 +2^{k+1}$</p> <p>$\Rightarrow$ $2.2^{k+1} -1$</p> <p>$\Rightarrow$ $2^{k+2}-1$</p> <p>$\therefore$ P(k+1) is true.</p> <p>Hence by mathematical induction</p> <p>P(n) is true for all $n \in \mathbb{N} \cup \{0\}$</p> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <ul> <li>Use M.I. to prove $2^n < n! \forall n \geq 4$</li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ <span style="color:blue">Example</span> $\to$ Solution $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <ul> <li>To P.T.</li> <li>P(n) is true for all $n \geq 4$</li> </ul> <p><strong>Basis step:</strong></p> <ul> <li>Verify P(4) is true</li> <li>$2^4 = 16 < 24=4!$</li> <li>n! = 1. 2. 3. ….n</li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <p><strong>Inductive step:</strong></p> <ul> <li> <p>Induction hypothesis is P(k) is true</p> </li> <li> <p>i.e. $2^k < k!$</p> </li> <li> <p>To prove P(k+1) is true</p> </li> <li> <p>i.e. to prove, $2^{k+1} < (k+1)!$</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <ul> <li> <p>$2^k < k!$</p> </li> <li> <p>For an arbitrary, but fixed k, where $k \geq 4$</p> </li> <li> <p>$2^{k+1} = 2.2^{k}< 2.k!$</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-1</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left; width:100%;font-size:22px;text-align: left'> <ul> <li> <p>$[(k+1)! = (k+1).k!]$</p> </li> <li> <p>$2^{k+1} < (k+1) k!=(k+1)!$</p> </li> <li> <p>$(\because 2 < k+1)$</p> </li> <li> <p>$\Rightarrow 2^{k+1} < (k+1)! $</p> </li> <li> <p>Hence, by principle of mathematical induction</p> </li> <li> <p>It follows that</p> </li> <li> <p>$2^{n} < n!$ fo all $n \geq 4$</p> </li> </ul> </div> </main> <hr> <footer style='font-size:18px'> A proof technique $\to$ Principle of mathematical induction $\to$ Remark $\to$ Induction hypothesis $\to$ Question $\to$ Example $\to$ <span style="color:blue">Solution</span> $\to$ Inductive step </footer>

<p><Success style='float: center;margin-top:16rem;text-align:center'><B>Thank you</B></Success></p>