1.2 Arithmetic sequences and series questions for IB Maths AI

36 questions on Arithmetic sequences and series, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 1 Number and algebra
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	19	0	19
Paper 2	15	1	16
Paper 3HL only	0	1	1
Every paper	34	2	36

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00029Paper 1 · 7 marks

The Teatro Alameda in Seville has 30 rows of seats. There are 22 seats in the first row, and each row after the first has 3 more seats than the row in front of it.

(a)
Find the number of seats in the 30th row.
2 marks
(b)
Find the total number of seats in the theatre.
2 marks
(c)
The theatre is to be extended by adding further rows at the back, continuing the same pattern, so that the theatre has at least 2500 seats.
Find the least number of rows that must be added.
3 marks

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Question 2 · AI-00187Paper 2 · 14 marks

An open-air theatre in Verona has rows of seats. Each row has the same number of extra seats as the row in front of it. Row 5 has 52 seats and the first 20 rows have 1370 seats altogether.

(a)
Let $u_{1}$ be the number of seats in row 1 and $d$ the number of extra seats in each row. Write down two equations in $u_{1}$ and $d$ .
2 marks
(b)
Find the value of $u_{1}$ and the value of $d$ .
2 marks
(c)
For a concert, at least 2500 seats are needed. The seats are used from row 1 backwards. Find the least number of rows that must be used.
3 marks
(d)
For this concert, a seat in row 1 costs 85 euros (EUR). The price in each row is 5% less than the price in the row in front of it. Find the price of a seat in row 12. Give your answer correct to two decimal places.
2 marks
(e)
Find how many rows have a seat price of at least EUR 50.
3 marks
(f)
The organiser says that a seat in row 30 costs less than a quarter of the price of a seat in row 1. Determine whether the organiser is correct.
2 marks

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Question 3 · AI-01333Paper 3 · HL only · 30 marks

In this question, give all monetary answers correct to two decimal places.

The purpose of this question is to compare three ways of repaying a business loan and to explore how an infinite geometric series can be used to value a business.

Ines Carvalho is opening a bakery in Porto. She needs to borrow 24 000 euros (EUR) to buy ovens and is offered three repayment options. In every option the payments are made at the end of each month for 5 years.

Option A is a standard loan with a nominal annual interest rate of 6.0 %, compounded monthly, repaid with equal monthly payments.

(a)
Find the monthly payment for Option A.
3 marks
(b)
Find the total amount of interest Ines pays with Option A.
2 marks
(c)
Find the amount Ines would still owe after 2 years with Option A.
2 marks
(d)(i)
Option B is advertised as a "flat-rate loan at only 4.2 % per year". For each year of the loan, interest equal to 4.2 % of the original amount borrowed is added to the debt, so the total debt after $n$ years, $u_{n}$ , forms an arithmetic sequence. The total debt after 5 years is then repaid in 60 equal monthly payments.
Show that the total debt after 5 years is EUR 29 040.
2 marks
(d)(ii)
Hence write down the monthly payment for Option B.
1 mark
(e)(i)
Find the nominal annual interest rate, compounded monthly, that Ines is actually being charged with Option B.
3 marks
(e)(ii)
Ines says that Option B is cheaper than Option A because 4.2 % is less than 6.0 %. Explain why she is wrong.
2 marks
(f)
Option C is a "growing payment" loan with the same interest rate as Option A, 6.0 % compounded monthly, which is 0.5 % per month. The first payment is EUR $P$ , and each later payment is 1 % larger than the previous one, so that the payments form a geometric sequence.
Write down an expression, in terms of $P$ and $k$ , for the $k$ th payment.
1 mark
(g)
The present value of a payment of $X$ made at the end of month $k$ is $\frac{X}{{1.005}^{k}}$ .
Show that the sum of the present values of all 60 payments is
$\frac{P}{1.005} \times \frac{q^{60} - 1}{q - 1}, where q = \frac{1.01}{1.005} .$
2 marks
(h)
The loan is fully repaid when the sum of the present values of the payments equals the amount borrowed.
Hence find the value of $P$ .
3 marks
(i)
Find
(i) the size of the final payment;
(ii) the total amount Ines repays with Option C.
3 marks
(j)
Option C costs more in total than Option A. Suggest one reason why Ines might still choose Option C.
1 mark
(k)
Ines later estimates the value of the bakery as the sum of the present values of all its future monthly profits, again discounting at 0.5 % per month. She models the profit at the end of the first month as EUR 1500, with each month's profit 0.3 % greater than the previous month's, and assumes this continues indefinitely.
Find the value of the bakery according to this model.
3 marks
(l)
Explain why this method of valuation could not be used if Ines assumed that the monthly profit grew by 0.6 % each month.
2 marks

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Question 4 · AI-00030Paper 1 · 6 marks

Kofi deposits 4500 Ghanaian cedi (GHS) in a savings account that pays simple interest at a rate of 3.8% per year. No further deposits or withdrawals are made.

The value of the account at the end of each year forms an arithmetic sequence.

(a)
Write down the common difference of this sequence.
1 mark
(b)
Find the value of the account at the end of the 12th year.
2 marks
(c)
Find the number of complete years until the value of the account first exceeds 7000 GHS.
3 marks

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Question 5 · AI-00343Paper 2 · 15 marks

The Halden Concert Hall has 24 rows of seats. Row 1, at the front, has 18 seats, and each row has 2 more seats than the row in front of it.

(a)(i)
Find the number of seats in row 24.
2 marks
(a)(ii)
Find the total number of seats in the hall.
2 marks
(b)
For a concert, a ticket in row 1 costs 640 Norwegian kroner (NOK), and the price of a ticket decreases by 15 NOK for each row further back.
Find the price of a ticket in row 24.
2 marks
(c)
Show that the income from selling every seat in row 10 is 18 180 NOK.
2 marks
(d)
Let $I (n)$ be the income, in NOK, from selling every seat in row $n$ .
Write down an expression for $I (n)$ in terms of $n$ .
2 marks
(e)
Find the total income when every seat in the hall is sold.
2 marks
(f)
Find which row produces the greatest income when all of its seats are sold, and write down this income.
3 marks

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Question 6 · AI-00031Paper 1 · 7 marks

In an arithmetic sequence, the fourth term is 23.5 and the eleventh term is 4.6.

(a)
Find the common difference.
2 marks
(b)
Find the first term.
1 mark
(c)
Find the sum of the first 20 terms.
2 marks
(d)
The $n$ th term of the sequence is $- 38.6$ . Find the value of $n$ .
2 marks

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