### sum geometric

Question Sample Titled 'sum geometric'

The angle of sector and the radius of the paper sector ${S}_{{1}}$ are ${80}^{\circ}$ and ${r}$ $\text{cm}$ respectively.

 (a) Find the area of ${S}_{{1}}$ in terms of $\pi$ and ${r}$ . (1 mark) (b) Paper sector ${S}_{{2}}$ is similar to ${S}_{{1}}$. It is given that the radius of ${S}_{{2}}$ is half of that of ${S}_{{1}}$. Find the area of ${S}_{{2}}$ in terms of $\pi$ and ${r}$ . (1 mark)

${S}_{{1}}$${S}_{{2}}$${S}_{{2}}$${S}_{{3}}$${S}_{{3}}$${S}_{{3}}$${S}_{{3}}$$\ldots$

 (c) Now on the table, stick two sectors ${S}_{{2}}$ on top of ${S}_{{1}}$. For each sector ${S}_{{2}}$, two more similar sectors ${S}_{{3}}$, in which the radius is half of that of ${S}_{{2}}$, are created and are sticked on top of the sector ${S}_{{2}}$, as shown in the figure. If the same process is repeated on each newly created sectors, Melissa thinks that the total area of all sectors except ${S}_{{1}}$ would be greater than the area of ${S}_{{1}}$. Do you agree? Explain your answer. (4 marks) The area at the back (i.e. facing the table) are not counted.

 (a) Required area $=\pi{r}^{{2}}\times\dfrac{{{80}^{\circ}}}{{{360}^{\circ}}}$  $=\dfrac{{2}}{{9}}\pi{r}^{{2}}$ $\text{cm}^{{2}}$ 1A (b) Required area $=\pi{\left(\dfrac{{r}}{{2}}\right)}^{{2}}\times\dfrac{{{80}^{\circ}}}{{{360}^{\circ}}}$  $=\dfrac{{1}}{{18}}\pi{r}^{{2}}$ $\text{cm}^{{2}}$ 1A (c) Total area of all sectors which are smaller than ${S}_{{1}}$ $=\dfrac{{2}}{{9}}\pi{r}^{{2}}{\left({2}{\left(\dfrac{{1}}{{2}}\right)}^{{2}}+{4}{\left(\dfrac{{1}}{{2}}\right)}^{{4}}+{8}{\left(\dfrac{{1}}{{2}}\right)}^{{6}}+\ldots\right)}$ 1M $=\dfrac{{2}}{{9}}\pi{r}^{{2}}{\left({\left(\dfrac{{1}}{{2}}\right)}^{{1}}+{\left(\dfrac{{1}}{{2}}\right)}^{{2}}+{\left(\dfrac{{1}}{{2}}\right)}^{{3}}+\ldots\right)}$ $=\dfrac{{2}}{{9}}\pi{r}^{{2}}\times\dfrac{{\dfrac{{1}}{{2}}}}{{{1}-\dfrac{{1}}{{2}}}}$ 1A $=\dfrac{{2}}{{9}}\pi{r}^{{2}}$ $\text{cm}^{{2}}$ 1A $=$Area of ${S}_{{1}}$ (Not greater than area of ${S}_{{1}}$) Thus, she is disagreed. 1A

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