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Date	May 2022	Marks available	3	Reference code	22M.2.SL.TZ2.2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Determine	Question number	2	Adapted from	N/A

Question

Scott purchases food for his dog in large bags and feeds the dog the same amount of dog food each day. The amount of dog food left in the bag at the end of each day can be modelled by an arithmetic sequence.

On a particular day, Scott opened a new bag of dog food and fed his dog. By the end of the third day there were $115.5$ cups of dog food remaining in the bag and at the end of the eighth day there were $108$ cups of dog food remaining in the bag.

Find the number of cups of dog food

In $2021$ , Scott spent $$ 625$ on dog food. Scott expects that the amount he spends on dog food will increase at an annual rate of $6.4 %$ .

fed to the dog per day.

[3]

a.i.

remaining in the bag at the end of the first day.

[1]

a.ii.

Calculate the number of days that Scott can feed his dog with one bag of food.

[2]

Determine the amount that Scott expects to spend on dog food in $2025$ . Round your answer to the nearest dollar.

[3]

Calculate the value of $Σ_{n = 1}^{10} (625 \times 1 . 064^{(n - 1)})$ .

[1]

d.i.

Describe what the value in part (d)(i) represents in this context.

[2]

d.ii.

Comment on the appropriateness of modelling this scenario with a geometric sequence.

[1]

Markscheme

EITHER

$115.5 = u_{1} + (3 - 1) \times d (115.5 = u_{1} + 2 d)$

$108 = u_{1} + (8 - 1) \times d (108 = u_{1} + 7 d)$ (M1)(A1)

Note: Award M1 for attempting to use the arithmetic sequence term formula, A1 for both equations correct. Working for M1 and A1 can be found in parts (i) or (ii).

$(d = - 1.5)$

$1.5$ (cups/day) A1

Note: Answer must be written as a positive value to award A1.

$(d =) \frac{115.5 - 108}{5}$ (M1)(A1)

Note: Award M1 for attempting a calculation using the difference between term $3$ and term $8$ ; A1 for a correct substitution.

$(d =) 1.5$ (cups/day) A1

[3 marks]

a.i.

$(u_{1} =) 118.5$ (cups) A1

[1 mark]

a.ii.

attempting to substitute their values into the term formula for arithmetic sequence equated to zero (M1)

$0 = 118.5 + (n - 1) \times (- 1.5)$

$(n =) 80$ days A1

Note: Follow through from part (a) only if their answer is positive.

[2 marks]

$(t_{5} =) 625 \times 1 . 064^{(5 - 1)}$ (M1)(A1)

Note: Award M1 for attempting to use the geometric sequence term formula; A1 for a correct substitution

$$ 801$ A1

Note: The answer must be rounded to a whole number to award the final A1.

[3 marks]

$(S_{10} =) ($) 8390 (8394.39 \dots)$ A1

[1 mark]

d.i.

EITHER

the total cost (of dog food) R1

for $10$ years beginning in $2021$ OR $10$ years before $2031$ R1

the total cost (of dog food) R1

from $2021$ to $2030$ (inclusive) OR from $2021$ to (the start of) $2031$ R1

[2 marks]

d.ii.

EITHER
According to the model, the cost of dog food per year will eventually be too high to keep a dog.

OR
The model does not necessarily consider changes in inflation rate.

OR
The model is appropriate as long as inflation increases at a similar rate.

OR
The model does not account for changes in the amount of food the dog eats as it ages/becomes ill/stops growing.

OR
The model is appropriate since dog food bags can only be bought in discrete quantities. R1

Note: Accept reasonable answers commenting on the appropriateness of the model for the specific scenario. There should be a reference to the given context. A reference to the geometric model must be clear: either “model” is mentioned specifically, or other mathematical terms such as “increasing” or “discrete quantities” are seen. Do not accept a contextual argument in isolation, e.g. “The dog will eventually die”.

[1 mark]

Examiners report

Parts (a) and (b) were mostly well answered, but some candidates ignored the context and did not give the number of dog food cups per day as a positive number. Most candidates considered geometric sequence in part (c) correctly, and used the correct formula for the nth term, although they used an incorrect value for n at times. Some candidates used the finance application incorrectly. The sum in part (d) was calculated correctly by some candidates, although many seemed unfamiliar with sigma notation and with calculating summations using GDC. In part (d), most candidates interpreted correctly that the sum represented the cost of dog food for 10 years but did not identify the specific 10-year period. Part (e) was not answered well – often candidates made very general and abstract statements devoid of any contextual references.

a.i.

[N/A]

a.ii.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

Syllabus sections

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

Show 111 related questions

18M.2.AHL.TZ1.H_1a:
Find the first term and the common difference of the sequence.
19M.2.AHL.TZ2.H_7:
Suppose that ${u_1}$ is the first term of a geometric series with common ratio $r$ .

Prove, by mathematical induction, that the sum of the first $n$ terms, ${s_n}$ is given by

${s_n} = \frac{{{u_1}\left( {1 - {r^n}} \right)}}{{1 - r}}$ , where $n \in {\mathbb{Z}^ + }$ .
22M.2.SL.TZ1.2c.ii:
Find the value of $w_{n}$ for this path.
22M.2.SL.TZ1.2b.ii:
$l_{n}$ .
22M.2.SL.TZ1.2c.i:
Show that Eddie needs $144$ tiles.
18M.2.AHL.TZ1.H_1b:
Calculate the number of positive terms in the sequence.
17N.2.SL.TZ0.T_2a.i:
Write down the distance Rosa runs in the third training session;
18M.1.SL.TZ1.T_7a.ii:
For that day find the weight of Sergei’s first lift.
16N.1.AHL.TZ0.H_6a:
Write down the value of ${u_1}$ .
SPM.1.SL.TZ0.2b:
Calculate the price of a ticket in the 16th row.
18M.2.SL.TZ2.T_4d:
Calculate the tea-shop’s total profit for the first 12 weeks.
22M.2.SL.TZ1.2a.ii:
$b$ .
22M.2.SL.TZ1.2b.i:
$w_{n}$ .
22M.2.SL.TZ2.2a.ii:
remaining in the bag at the end of the first day.
16N.1.SL.TZ0.S_9d:
Show that ${S_{12}} = 12{\log _2}x - 66$ .
18M.1.AHL.TZ2.H_5a:
Show that A is an arithmetic sequence, stating its common difference d in terms of r.
22M.2.SL.TZ2.2b:
Calculate the number of days that Scott can feed his dog with one bag of food.
17N.1.SL.TZ0.S_2b:
Find the tenth term.
21N.2.SL.TZ0.2f:
State whether, for all $n > k$ , the university will have places available for all applicants. Justify your answer.
22M.2.SL.TZ2.2a.i:
fed to the dog per day.
22M.2.SL.TZ1.2a.i:
$a$ .
21N.2.SL.TZ0.2b.i:
Write down the common ratio of the sequence.
17M.1.AHL.TZ2.H_3b:
the value of $r$ ;
16N.1.SL.TZ0.S_9b:
Show that the sum of the infinite sequence is $4{\log _2}x$ .
19M.1.AHL.TZ2.H_1:
In an arithmetic sequence, the sum of the 3rd and 8th terms is 1.

Given that the sum of the first seven terms is 35, determine the first term and the common difference.
18M.2.SL.TZ2.T_4c:
Find the tea-shop’s profit during the 11th week.
18M.2.SL.TZ2.T_4e:
In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.

Find the value of m.
17M.1.AHL.TZ1.H_7b:
determine the value of $\sum\limits_{r = 1}^N {{u_r}}$ .
17M.1.SL.TZ2.T_5b:
Find the total number of sticks used by Tomás for all 24 diagrams.
18M.2.SL.TZ2.T_4b:
Calculate the café’s total profit for the first 12 weeks.
16N.2.AHL.TZ0.H_12a:
Find an expression for ${A_1}$ and show that ${A_2} = {1.004^2}x + 1.004x$ .
18M.1.SL.TZ1.T_7a.i:
For that day find how much weight was added after each lift.
19M.2.SL.TZ2.T_4d.iii:
Hence find the total amount of antibiotic, in mg, that Ted receives during the first $k$ days.
19M.2.SL.TZ2.T_4b:
Write down the value of $d$ and the value of ${u_1}$ .
17N.2.SL.TZ0.T_2a.ii:
Write down the distance Rosa runs in the $n$ th training session.
16N.1.SL.TZ0.S_9a:
Find $r$ .
18M.2.SL.TZ1.S_7a:
Given that x_k_{+ 1} = x_k + a, find a.
19M.2.SL.TZ1.T_5c:
Calculate, in years, when the bicycle value will be less than 50 USD.
17N.2.SL.TZ0.T_2b:
Find the value of $k$ .
SPM.1.SL.TZ0.2a:
Write down the value of the common difference, $d$
19M.2.SL.TZ1.T_5a:
Calculate, in CAD, the total amount John pays for the bicycle.
17N.1.SL.TZ0.S_2c:
Find the sum of the first ten terms.
16N.1.AHL.TZ0.H_6b:
Find the value of ${u_6}$ .
17N.2.SL.TZ0.T_2d:
Find the distance Carlos runs in the fifth month of training.
17M.1.AHL.TZ1.H_7a:
find the value of $d$ .
17M.1.SL.TZ1.T_5b:
The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.
16N.1.AHL.TZ0.H_6c:
Prove that $\{ {u_n}\}$ is an arithmetic sequence, stating clearly its common difference.
18N.2.SL.TZ0.S_5:
The sum of an infinite geometric sequence is 33.25. The second term of the sequence is 7.98. Find the possible values of $r$ .
19N.2.SL.TZ0.T_3b:
Show that the total number of triangular panes, $S_{n}$ , in the first $n$ levels is given by:

$S_{n} = n^{2} + 4 n$ .
17N.1.SL.TZ0.S_2a:
Find the common difference.
19M.2.SL.TZ2.T_4a.i:
Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the seventh day.
18N.1.SL.TZ0.S_3a:
Find u₈.
16N.1.SL.TZ0.S_9e:
Given that ${S_{12}}$ is equal to half the sum of the infinite geometric sequence, find $x$ , giving your answer in the form ${2^p}$ , where $p \in \mathbb{Q}$ .
16N.2.AHL.TZ0.H_12e:
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.
18M.1.SL.TZ1.T_7b:
On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.
16N.2.AHL.TZ0.H_12d:
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.
20N.2.SL.TZ0.T_5c:
Write down the common difference, $d$ .
20N.2.SL.TZ0.T_5f:
Find the total amount of fuel pumped into the tank in the first $8$ hours.
20N.2.SL.TZ0.T_5g:
Show that the tank will never be completely filled using this pump.
19M.2.SL.TZ2.T_4d.ii:
The daily amount of antibiotic Ted receives will first be less than 0.06 mg on the $k$ th day. Find the value of $k$ .
19N.2.SL.TZ0.T_3e:
Each triangular pane has an area of $1.84 m^{2}$ .

Find the total area of the decorative glass face, if the maximum number of complete levels were built. Express your area to the nearest $m^{2}$ .
20N.2.SL.TZ0.T_5e.ii:
Write down the number of hours that the pump was pumping fuel into the tank.
EXN.1.SL.TZ0.12a:
Write down the value of $Σ_{i = 1}^{9} u_{i}$ .
EXN.1.SL.TZ0.12b:
Find the value of $u_{1}$ .
19M.2.SL.TZ2.T_4d.i:
Find the amount of antibiotic, in mg, that Ted receives on the fifth day.
18M.2.SL.TZ2.T_4a:
Find the café’s profit during the 11th week.
19M.2.SL.TZ2.S_10b:
Find the exact value of ${S_k}$ .
17N.2.SL.TZ0.T_2e:
Calculate the total distance Carlos runs in the first year.
18M.2.AHL.TZ1.H_7a:
Show that there will be approximately 2645 fish in the lake at the start of 2020.
16N.1.SL.TZ0.S_9c:
Find $d$ , giving your answer as an integer.
19N.2.SL.TZ0.T_3c:
Hence, find the total number of panes in a glass face with $18$ levels.
16N.2.AHL.TZ0.H_12b:
(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 $251({1.004^{120}} - 1)x$ .
18M.2.SL.TZ1.S_7b:
Hence find the value of n such that $\sum\limits_{k = 1}^n {{x_k} = 861}$ .
19M.2.SL.TZ2.T_4c:
Calculate the total amount of the drug, in mg, that she receives.
20N.2.SL.TZ0.T_5b:
Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.
19M.2.SL.TZ2.S_10d:
An infinite geometric series is given as ${S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots$ , $a \in {\mathbb{Z}^ + }$ .

Find the largest value of $a$ such that ${S_\infty } < F$ .
18M.2.AHL.TZ1.H_7b:
Find the approximate number of fish in the lake at the start of 2042.
19M.2.SL.TZ1.S_7a:
Write down the first three non-zero terms of ${w_n}$ .
20N.2.SL.TZ0.T_5a:
Find $h$ , the height of the tank.
17M.1.SL.TZ2.T_5a:
Diagram $n$ is formed with 52 sticks. Find the value of $n$ .
21M.1.SL.TZ1.8a.i:
Charlie ran on day $20$ of his fitness programme.
20N.2.SL.TZ0.T_5d:
Find the amount of fuel pumped into the tank in the $13th$ hour.
19N.2.SL.TZ0.T_3a:
Find the number of triangular panes in the $12 th$ level.
SPM.1.SL.TZ0.2c:
Find the total cost of buying 2 tickets in each of the first 16 rows.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
19N.2.SL.TZ0.T_3d:
Find the maximum number of complete levels that Maegan can build.
18M.1.SL.TZ2.S_7b:
Let $p = {c^2}$ and $q = {c^3}$ . Find the value of $\sum\limits_{n = 1}^{20} {{u_n}}$ .
21M.2.SL.TZ2.3a.ii:
the total number of seats in the concert hall.
21M.2.SL.TZ2.3a.i:
the number of seats in the last row.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
19M.2.SL.TZ2.T_4a.ii:
Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the eleventh day.
18M.1.AHL.TZ2.H_5b:
A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.
19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
19M.2.SL.TZ2.S_10c:
Show that $F = 3240$ .
19M.2.SL.TZ1.T_5d:
Find the total amount John has paid to insure his bicycle for the first 5 years.
20N.2.SL.TZ0.T_5e.i:
Find the value of $n$ such that $u_{n} = 0$ .
16N.2.AHL.TZ0.H_12c:
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 $n$ .
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
EXN.3.AHL.TZ0.2a:
Find an expression for the width of $U_{n}$ in centimetres.
19M.2.SL.TZ1.T_5b:
Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
EXN.1.SL.TZ0.12c:
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 $k$ .
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
Daniella ran on day $20$ of her fitness programme.
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:
Calculate the percentage increase in applications from the first year to the second year.
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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