| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Arithmetic/Geometric Series Applications |
| Difficulty | Standard +0.8 This question requires understanding of both arithmetic and geometric series, calculating 5% of each term, and summing these donations over 40 days. While the individual concepts are standard A-level material, the multi-step nature (identifying series type, finding the donation formula, applying summation formulas for both models) and the need to work with percentages of series terms elevates this beyond a routine exercise. The geometric series calculation with the 1.1 multiplier and careful arithmetic pushes this slightly above average difficulty. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<11.04k Modelling with sequences: compound interest, growth/decay |
| Answer | Marks | Guidance |
|---|---|---|
| (i) 1000, 2000, 3000... or 50, 100, 150... | M1 | Recognise series, correct a/d (or 3 terms ) |
| \(\frac{40}{2(1000 + 400000)}\) or \(\frac{40}{2(2000 + 390000)}\) | M1 | Correct use of formula |
| \(\times 5\%\) of attempt at valid sum 41000 | M1 | |
| A1 [4] | Can be awarded in either (i) or (ii) cao | |
| (ii) \(1000, 1000 \times 1.1, 1000 \times 1.1^2 + ...\) or with \(a = 50\) \(1000(1.1^{40} - 1)\) | M1 | Recognise series, correct a/r (or 3 terms) |
| M1 | Correct use of formula. Allow e.g. r = 0.1 | |
| \(\frac{1.1 - 1}{22100}\) | A1 [3] | Or answers rounding to this |
**(i)** 1000, 2000, 3000... or 50, 100, 150... | M1 | Recognise series, correct a/d (or 3 terms )
$\frac{40}{2(1000 + 400000)}$ or $\frac{40}{2(2000 + 390000)}$ | M1 | Correct use of formula
$\times 5\%$ of attempt at valid sum 41000 | M1 |
| A1 [4] | Can be awarded in either (i) or (ii) cao
**(ii)** $1000, 1000 \times 1.1, 1000 \times 1.1^2 + ...$ or with $a = 50$ $1000(1.1^{40} - 1)$ | M1 | Recognise series, correct a/r (or 3 terms)
| M1 | Correct use of formula. Allow e.g. r = 0.1
$\frac{1.1 - 1}{22100}$ | A1 [3] | Or answers rounding to this
---8 A television quiz show takes place every day. On day 1 the prize money is $\$ 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money.
Model 1: Increase the prize money by $\$ 1000$ each day.\\
Model 2: Increase the prize money by $10 \%$ each day.\\
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is $5 \%$ of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity\\
(i) if Model 1 is used,\\
(ii) if Model 2 is used.
\hfill \mbox{\textit{CAIE P1 2011 Q8 [7]}}