Question
Using principle of mathematical induction prove that $\sqrt{\text{n}}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{n}}}$ for all natural numbers $\text{n}\geq2.$
Step 1:
p(2): $\sqrt{2}=1.4142$ $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}=1+\frac{1}{1.4142}=1+0.7071=1.7071$ $\therefore\sqrt{2}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}$ $\therefore$ p(2) is true.Step 2:
Let p(m) is true. Then, $\sqrt{\text{m}}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}} \ ...(1)$ We have to prove that p(m + 1) is true. $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}}>\sqrt{\text{m}} \ ...[\text{From}(1)]$ $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}}+\frac{1}{\sqrt{\text{m}+1}}>\sqrt{\text{m}}+\frac{1}{\sqrt{\text{m}+1}}$ $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}}+\frac{1}{\sqrt{\text{m}+1}}>\frac{\sqrt{\text{m}^2+\text{m}}+1}{\sqrt{\text{m}+1}}$ $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}}+\frac{1}{\sqrt{\text{m}+1}}>\frac{\sqrt{\text{m}^2}+1}{\sqrt{\text{m}+1}}$ $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}}+\frac{1}{\sqrt{\text{m}+1}}>\frac{\text{m}+1}{\sqrt{\text{m}+1}}$ $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{\text{m}}}+\frac{1}{\sqrt{\text{m}+1}}>\sqrt{\text{m}+1}$ ⇒ p(m + 1) is true. Hence by the principle of mathematical induction, the given results is true for all $\text{n}\in\text{N}.$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| Column C1 | Column C2 | ||
| (a) | $\sin(\text{x + y})\sin\text{x}-\text{y}$ | (i) | $\cos^2\text{x}-\sin^2\text{y}$ |
| (b) | $\cos(\text{x + y})\cos(\text{x}-\text{y})$ | (ii) | $\frac{1-\tan\theta}{1+\tan\theta}$ |
| (c) | $\cot\Big(\frac{\pi}{4}+\theta\Big)$ | (iii) | $\frac{1+\tan\theta}{1-\tan\theta}$ |
| (d) | $\tan\Big(\frac{\pi}{4}+\theta\Big)$ | (iv) | $\sin^2\text{x}-\sin^2\text{y}$ |