APGP Question #3

This is a question from a recent ACJC JC2 test.

A philanthropist started a donation matching programme to encourage more people to donate regularly to a particular charity.

(a) For a person who donates <em>a</em> in the first month, and for each subsequent month donates b^2 more than the previous month, the philanthropist will donate <em>a</em> in the first month and for each subsequent month, <em>b</em> times the amount he donated the previous month.  (i) John is a regular donor of the charity. Find the values of <em>a</em> and <em>b</em> such that the philanthropist donates20 in the third month, and ten times more than John in the seventh month.

(ii) Find the total amount of money donated by John and the philanthropist in one year, leaving your answer to the nearest dollar.

(b) In a revised donation matching programme, if a donor makes a monthly donation of <em>c</em>, the philanthropist will donate according to the following plan:  First month : No donation (latex 0\% donation rate) Second month :latex 10 \% of the total amount donated by the donor in the first two months Third month :latex 20 \% of the total amount donated by the donor in the first three months Fourth month :latex 30 \% of the total amount donated by the donor in the first four months  and so on.  (i) Show that the total amount of money the philanthropist will donate at the end of thelatex 4^{th} month is2c.

(ii) By expressing the total amount of money, the philanthropist will donate at the end of the n^{th} month as a summation, and using the result \sum_{r=1}^n r^2 = \frac{n}{6}(n+1)(2n+1), show that the total amount of money the philanthropist will donate at the end of the n^{th} month is latex \frac{(n-1)n(n+1)}{30} c .  [showhide type="Solution" more_text="Solution" less_text="Hide Solution"]  (i) <table cellspacing="0" cellpadding="0"> <tbody> <tr> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex n^{th}month</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">John</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">Philanthropist </span></td> </tr> <tr> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">1</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex a</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex a</span></td> </tr> <tr> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">2</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex a + b^2 </span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex ab</span></td> </tr> <tr> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">...</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">...</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">...</span></td> </tr> <tr> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">7</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex a + 6b^2</span></td> <td valign="top"><span style="color: #000000; font-family: Helvetica; font-size: small;">latex ab^6</span></td> </tr> <tr> <td valign="top"></td> <td valign="top"></td> <td valign="top"></td> </tr> </tbody> </table>latex ab^2 = 20 ---(1)latex ab^6 = 10(a+ 6b^2) ---(2)  (1):latex b^2 = \frac{20}{a}latex \Rightarrow a(\frac{20}{a})^3 = 10[a + 6(\frac{20}{a})]latex a^3 + 120a – 800 = 0Using GC,latex a = 5.373610646 \approx 5.37 (3 SF)latex b = 1.929220642 \approx 1.93 (3 SF)(ii) Amount John donatedlatex = \frac{12}{2} [2(5.373610646) + (12-1)(1.929220642)^2] = 310.1282186 Amount Philanthropist donatedlatex = \frac{5.373610646(1.929220642^{12} – 1)}{1.929220642 – 1} = 15366.20901 Total Amountlatex = 15366.20901 + 310.1282186 = 15676.33723 \approx 15676.34$

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