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Date	November 2016	Marks available	1	Reference code	16N.2.AHL.TZ0.H_12
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Write down	Question number	H_12	Adapted from	N/A

Question

On the day of her birth, 1st January 1998, Mary’s grandparents invested $\$ x$ in a savings account. They continued to deposit $\$ x$ on the first day of each month thereafter.

The account paid a fixed rate of 0.4% interest per month. The interest was calculated on the last day of each month and added to the account.

Let $\$ {A_n}$ be the amount in Mary’s account on the last day of the $n{\text{th}}$ month, immediately after the interest had been added.

Find an expression for ${A_1}$ and show that ${A_2} = {1.004^2}x + 1.004x$ .

[2]

(i) Write down a similar expression for ${A_3}$ and ${A_4}$ .

(ii) Hence show that the amount in Mary’s account the day before she turned 10 years old is given by $251({1.004^{120}} - 1)x$ .

[6]

Write down an expression for ${A_n}$ in terms of $x$ on the day before Mary turned 18 years old showing clearly the value of $n$ .

[1]

Mary’s grandparents wished for the amount in her account to be at least $\$ 20\,000$ the day before she was 18. Determine the minimum value of the monthly deposit $\$ x$ required to achieve this. Give your answer correct to the nearest dollar.

[4]

As soon as Mary was 18 she decided to invest $\$ 15\,000$ of this money in an account of the same type earning 0.4% interest per month. She withdraws $\$ 1000$ every year on her birthday to buy herself a present. Determine how long it will take until there is no money in the account.

[5]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

${A_1} = 1.004x$ A1

${A_2} = 1.004(1.004x + x)$ A1

$= {1.004^2}x + 1.004x$ AG

Note: Accept an argument in words for example, first deposit has been in for two months and second deposit has been in for one month.

[2 marks]

(i) ${A_3} = 1.004({1.004^2}x + 1.004x + x) = {1.004^3}x + {1.004^2}x + 1.004x$ (M1)A1

${A_4} = {1.004^4}x + {1.004^3}x + {1.004^2}x + 1.004x$ A1

(ii) ${A_{120}} = ({1.004^{120}} + {1.004^{119}} + \ldots + 1.004)x$ (A1)

$= \frac{{{{1.004}^{120}} - 1}}{{1.004 - 1}} \times 1.004x$ M1A1

$= 251({1.004^{120}} - 1)x$ AG

[6 marks]

${A_{216}} = 251({1.004^{216}} - 1)x{\text{ }}\left( { = x\sum\limits_{t = 1}^{216} {{{1.004}^t}} } \right)$ A1

[1 mark]

$251({1.004^{216}} - 1)x = 20\,000 \Rightarrow x = 58.22 \ldots$ (A1)(M1)(A1)

Note: Award (A1) for $251({1.004^{216}} - 1)x > 20\,000$ , (M1) for attempting to solve and (A1) for $x > 58.22 \ldots$ .

$x = 59$ A1

Note: Accept $x = 58$ . Accept $x \geqslant 59$ .

[4 marks]

$r = {1.004^{12}}{\text{ }}( = 1.049 \ldots )$ (M1)

$15\,000{r^n} - 1000\frac{{{r^n} - 1}}{{r - 1}} = 0 \Rightarrow n = 27.8 \ldots$ (A1)(M1)(A1)

Note: Award (A1) for the equation (with their value of $r$ ), (M1) for attempting to solve for $n$ and (A1) for $n = 27.8 \ldots$

$n = 28$ A1

Note: Accept $n = 27$ .

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

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Write down the first three non-zero terms of ${w_n}$ .
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Find the number of triangular panes in the $12 th$ level.
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Find the distance from the base of this ladder to the top rung.
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21M.2.SL.TZ2.3a.i:
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17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
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Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
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A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
19M.2.SL.TZ2.S_10a:
Find the value of $k$ .
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Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
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EXN.3.AHL.TZ0.2b:
Given the width of a pixel is approximately $0.025 cm$ , find the number of squares in the final image.
21N.2.SL.TZ0.2a:
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21N.2.SL.TZ0.2b.ii:
Find an expression for $u_{n}$ .
21N.2.SL.TZ0.2b.iii:
Find the number of student applications the university expects to receive when $n = 11$ . Express your answer to the nearest integer.
21N.2.SL.TZ0.2c:
Write down an expression for $v_{n}$ .
21N.2.SL.TZ0.2d:
Calculate the total amount of acceptance fees paid to the university in the first $10$ years.
21N.2.SL.TZ0.2e:
Find $k$ .

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Topic 1—Number and algebra

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