Question

Show that,

${\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{100} } < 20}$


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Answer 1

Step-by-step explanation:

I hope that this sum will help ur question

Answer 2

$\begin{gathered}\rm \pink{We \: have \: to \: prove \: that,}\end{gathered}\\ \begin{gathered}\rm\pink{ {\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{100} } < 20}}\end{gathered}\\ \begin{gathered}\rm \pink{To \: prove \: this \: result, let's \: prove \: first,}\end{gathered}\\\begin{gathered}\rm \pink{{\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{n} } < 2 \sqrt{n}}}\end{gathered}\\\begin{gathered}\rm \pink{where, n \: is \: a \: natural \: number}\end{gathered}\\ \begin{gathered}\rm \pink{To \: prove \: this \: result, we \: use \: Principal \: of \: Mathematical \: Induction.}\end{gathered} \\\begin{gathered}\rm \pink{Let \: assume \: that}\end{gathered} \\\begin{gathered}\rm \pink{P(n) : {\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{n} } < 2 \sqrt{n}}} \end{gathered}\\ \begin{gathered}\rm \pink{Step :- 1 \: For \: n = 1}\end{gathered}\\\begin{gathered}\rm \pink{P(1) : 1 < 2}\end{gathered}\\\begin{gathered}\rm \pink{{\implies \:P(n) \: is \: true \: for \: n \: = \: 1}}\end{gathered}\\\begin{gathered}\rm \pink{Step :- 2 \: Assume \: that \: P \: (n) \: is \: true \: for \: n \: = k, where \: k \: is \: some \: natural \: number.}\end{gathered}\\\begin{gathered}\rm \pink{P(k) : {\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{k} } < 2 \sqrt{k} } - - - (1)} \end{gathered}\\\begin{gathered}\rm \pink{Step :- 3 \: We \: have \: to \: prove \: that \: P \: (n) \: is \: true \: for \: n \: = k + 1}\end{gathered} \\\begin{gathered}\rm \pink{P(k + 1):{\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{k}} + \frac{1}{ \sqrt{k + 1} } < 2 \sqrt{k + 1}}}\end{gathered}\\\begin{gathered}\rm\pink{From \: equation \: (1), we \: have}\end{gathered}\\\begin{gathered}\rm\pink{{\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{k} } < 2 \sqrt{k}}} \end{gathered}\\\begin{gathered}\rm \pink{can \: be \: further \: rewritten \: as}\end{gathered}\\\begin{gathered}\rm \pink{{\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{k} } + \frac{1}{ \sqrt{k + 1} } < 2 \sqrt{k} } + \frac{1}{\sqrt{k + 1}}}\end{gathered} \\\begin{gathered}\rm \pink{< 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} + 2 \sqrt{k} - 2 \sqrt{k + 1}} \end{gathered}\\\begin{gathered}\rm \pink{< 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} + 2 (\sqrt{k} - \sqrt{k + 1})} \end{gathered}\\\begin{gathered}\rm \pink{ < 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} + 2 \bigg(\sqrt{k} - \sqrt{k + 1} \times \frac{ \sqrt{k} + \sqrt{k + 1}}{ \sqrt{k} + \sqrt{k + 1}} \bigg)} \end{gathered}\\\begin{gathered}\rm \pink{< 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} + 2 \bigg( \frac{k - k - 1}{ \sqrt{k} + \sqrt{k + 1}} \bigg)} \end{gathered}\\\begin{gathered}\rm \pink{< 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} - 2 \bigg( \frac{1}{ \sqrt{k} + \sqrt{k + 1}} \bigg)}\end{gathered} \\\begin{gathered}\rm \pink{ < 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} - 2 \bigg( \frac{1}{ \sqrt{k + 1} + \sqrt{k + 1}} \bigg)} \end{gathered}\\\begin{gathered}\rm \pink{< 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} - 2 \bigg( \frac{1}{ \sqrt{k + 1} + \sqrt{k + 1}} \bigg)}\end{gathered}\\\begin{gathered}\rm \pink{ < 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} - 2 \bigg( \frac{1}{2 \sqrt{k + 1}} \bigg)}\end{gathered}\\\begin{gathered}\rm \pink{ < 2 \sqrt{k} + \frac{1}{\sqrt{k + 1}} - \bigg( \frac{1}{\sqrt{k + 1}} \bigg)}\end{gathered}\\\begin{gathered}\rm \pink {< 2 \sqrt{k}}\end{gathered}\\\begin{gathered}\rm \pink{< 2 \sqrt{k + 1}}\end{gathered}\\\begin{gathered}\rm\pink{Hence,}\end{gathered}\\\begin{gathered}\rm\pink{{\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{k}} + \frac{1}{ \sqrt{k + 1} } < 2 \sqrt{k + 1}}} \end{gathered}\\\begin{gathered}\rm \pink{\implies \:P(n) \: is \: true \: for \: n \: = \: k + 1} \end{gathered} \\\begin{gathered}\rm\pink{Hence, By \: the \: Process \: of \: Principal \: of \: Mathematical \: Induction, we \: get}\end{gathered}\\\begin{gathered}\rm \pink{{\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{n} } < 2 \sqrt{n}}} \end{gathered}\\\begin{gathered}\rm \pink{Now, Substitute \: n = 100, in \: this \: result, we \: get}\end{gathered}\\\begin{gathered}\rm \pink{ {\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{100} } < 2 \sqrt{100}}} \end{gathered}\\\begin{gathered}\rm \pink{\implies \:\rm \: {\displaystyle\frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{100} } < 20}} \end{gathered}$