Hi,

I need help with the attached question! I need help with Part iii of Q7.

Thanks!

\[ \text{We know that}\\ \begin{align*} \frac{1}{2\times 2} &< \frac{1}{1\times 2}\\ \frac{1}{3\times 3}&< \frac{1}{2\times 3}\\ \frac{1}{4\times 4} &< \frac{1}{3\times 4}\\ &\vdots\\ \frac{1}{98\times 98} &< \frac{1}{97\times 98}\\ \frac{1}{99\times 99}&< \frac{1}{98\times 99} \end{align*} \]

\[ \text{So upon adding we have}\\ \begin{align*}&\quad \frac{1}{2^2} +\frac{1}{3^2} + \frac{1}{4^2} + \dots + \frac{1}{98^2}+\frac{1}{99^2}\\ & < \frac{1}{1\times 2} + \frac{1}{2\times 3} + \frac{1}{3\times 4} + \dots + \frac{1}{97\times 98} + \frac{1}{98\times 99}\\ &= \frac{99}{100} - \frac{1}{99\times 100} \tag{result from ii)}\\ &< \frac{99}{100} \end{align*}\\ \text{as required.} \]

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\[ \text{We also know that}\\ \begin{align*} \frac{1}{2\times 2} &> \frac{1}{2\times 3}\\ \frac{1}{3\times 3}&> \frac{1}{3\times 4}\\ &\vdots\\ \frac{1}{99\times 99} &> \frac{1}{99\times 100} \end{align*} \]

\[ \text{So upon adding we have}\\ \begin{align*}&\quad \frac{1}{2\times 2}+ \frac{1}{3\times 3} + \dots + \frac{1}{99\times 99}\\ &> \frac{1}{2\times 3} + \frac{1}{3\times 4} + \dots + \frac{1}{99\times 100}\\ &+ \frac{99}{100} - \frac{1}{1\times 2} \tag{result from ii)}\\ &= \frac{49}{100} \end{align*}\\ \text{as required.}\]