mathematics-class-11-unit-04-chapter-02-mathematical-induction-l-2-2-tqkn06lvnya

<p><Success style='float: center;margin-top:16rem;text-align:center'><B>Mathematical induction lecture-2</B></Success></p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Recall: principle of mathematical induction</B></Primary></p>
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<p>P(n) : Assertion involving ’n'</p>
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<p>Goal: To prove P(n) is true for all $n \in \mathbb{N}$</p>
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<p>This can be achieved in two steps:</p>
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<p><strong>Basis step</strong></p>
<p>To verify P(1) is true.</p>
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<span style="color:blue">Recall: principle of mathematical induction</span> $\to$ Recall $\to$ Harmonic number $\to$ Example $\to$ Proof $\to$  Harmonic series $\to$ Theorem $\to$ Remark
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p>
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<p>Prove that: $ H_{2^{n}} \geq 1 + \frac{n}{2} $ $ \forall n \in \N \cup \{0\} . $</p>
<p><strong>Proof:</strong></p>
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<p>Basis step</p>
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<p>P(n)$:H_{2^{n}} \geq 1 + \frac{n}{2}$</p>
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<p>To prove P(0)</p>
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<p>$H_{2^{0}} = H_1 = 1 \geq 1 + \frac{0}{2}$</p>
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<p>i.e P(0) is true.</p>
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<footer style='font-size:18px'>
Recall: principle of mathematical induction $\to$ Recall $\to$ Harmonic number $\to$ <span style="color:blue">Example</span> $\to$ Proof $\to$  Harmonic series $\to$ Theorem $\to$ Remark
</footer>
<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Mathematical induction lecture-2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Proof</B></Primary></p>
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<p>Inductive step</p>
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<p>Assume that P(k) is true.</p>
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<p>$H_{2^{k}} = 1+ \frac{1}{2}+ …. + \frac{1}{2^k} \\ \quad \quad \geq 1+ \frac{k}{2}$</p>
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<p>$\text{To prove that P(k+1) is true}$.</p>
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<p>$H_{2^{k+1}} = 1+ \frac{1}{2} + …. + \frac{1}{2^{k+1}}$</p>
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<p>$H_{2^{k+1}} =\biggl(1+ \frac{1}{2}+…+\frac{1}{2^{k}}\biggl)+\frac{1}{2^{k}+1}+…+\frac{1}{2^{k+1}}$</p>
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<hr>
<footer style='font-size:18px'>
Recall: principle of mathematical induction $\to$ Recall $\to$ Harmonic number $\to$ Example $\to$ <span style="color:blue">Proof</span> $\to$  Harmonic series $\to$ Theorem $\to$ Remark
</footer>