High School Maths • Grade 12 • Unit 1 • Review of Patterns and Sequences
Patterns Are Everywhere: Reviewing Arithmetic and Geometric Sequences
From the cloud patterns of Mega Mendung batik in Cirebon to the Islamic calendar, see number patterns with new eyes.
AMI | Global Mathematics Education | www.ami.sch.id
📚 Contents (tap to open)
- Learning Map
- Two Strange Questions
- Today's Mission
- 2-Minute Warm-Up
- Meet Again: Patterns, Sequences, Terms
- Arithmetic Sequences
- Geometric Sequences
- Arithmetic or Geometric?
- Next Level: Quadratic Sequences
- Shortcut: Relating Any Two Terms
- Case Study: Panjang Jimat and the Islamic Calendar
- One Question, Three Styles
- Watch Out for Traps!
- Mini Dictionary
- Your Turn!
- 30-Second Cheat Sheet
- Closing Quiz
- Quick Reflection
- Answer Key
- References
🧭Learning Map: 4 Lenses, 1 Topic
One topic, four ways to look at it. You will learn the same maths using the language of school, of international exams, of Islamic boarding schools, and of Cirebon culture (Cirebon is a city on the north coast of Java, Indonesia).
| Lens | What we use in this series | Source |
|---|---|---|
| Indonesian national curriculum | Phase F, Number strand: explain sequences and series (arithmetic and geometric) and apply them. This series reviews the foundations. | BSKAP Decree No. 046/H/KR/2025 |
| Cambridge and IB | The words progression, the symbols un, a, d, r; the “Series” topic in Pure Mathematics 1; the sequences and series topic in IB. | Cambridge 9709 syllabus (2026–2027); IB DP Mathematics |
| Islamic boarding school (pesantren) | Ilmu Falak (hisab): the science of calculating Islamic months and years. The value of istiqamah (being consistent) as an idea for a constant difference. | NU Online; falak studies at UIN Walisongo |
| Cirebon culture | Mega Mendung batik (layered clouds with graded colours) and the Panjang Jimat tradition (the Prophet's birthday celebration in the Cirebon palaces). | Iconography studies and university theses |
Note: there is no single standard “NU maths textbook”. So we connect through two things that are truly alive in Islamic boarding schools: Ilmu Falak (hisab) and the value of istiqamah. (NU is Nahdlatul Ulama, a large Indonesian Muslim organisation.)
1Two Strange Questions
Question 1. Look at Mega Mendung batik cloth from Cirebon. The clouds are layered, with colours that fade step by step. Why does it feel like the layers “grow” in a regular way?
Question 2. The Prophet Muhammad's birthday (Maulid Nabi) is on 12 Rabiul Awal. Why does its date in the normal (Gregorian) calendar move about 11 days earlier every year, and why can we predict it long before it happens?
Both answers are the same: there is a number pattern we can write as a formula. That is the job of sequences.
Mega Mendung is a batik pattern from Cirebon that shows layered clouds. The colours go from dark to light, and the cloud shapes are linked to Chinese cultural influence on Cirebon's coastal batik. The colour steps are often shown as about seven layers. A famous centre for batik makers is the Trusmi area.
Our own illustration, inspired by the Mega Mendung cloud layers. It is not a copy of the real pattern. The widths 2.0 cm and the extra 0.5 cm are our model, not a standard size used by batik makers.
2Today's Mission
After reading this lesson, you can:
- recognise arithmetic and geometric sequences from their first few terms;
- write and use the formula for the nth term, and find missing terms;
- explain how sequences connect to linear functions and exponential functions from Grade 10;
- read the same question in national, Cambridge, and IB styles;
- build a simple model from real data (the Islamic calendar) and judge its limits.
32-Minute Warm-Up
a. 5, 8, 11, 14, ... (next term?)
b. 3, 6, 12, 24, ... (next term?)
c. 1, 4, 9, 16, ... (next term?)
d. f(x) = 3x + 2. Find f(1), f(2), f(3). What pattern do you see?
The answers are in the Answer Key at the end. If you missed two or more, do not worry. This lesson is a review.
4Meet Again: Patterns, Sequences, Terms
A sequence is a list of numbers that follows a rule. Each number in the list is called a term. The first term is written u1 (or a), the second term u2, and so on up to the nth term un.
Example: 2, 2.5, 3, 3.5, 4, ... is the sequence of our cloud-layer widths. Here u1 = 2 and u2 = 2.5.
| Word | Symbol | Indonesian word |
|---|---|---|
| nth term | un (Indonesian books: Un) | suku ke-n |
| first term | a (or u1) | suku pertama |
| common difference | d (Indonesian books: b) | beda |
| common ratio | r | rasio |
| sequence / progression | – | barisan |
Indonesian textbooks usually write Un and b. Cambridge and IB write un and d. The meaning is the same, only the letters differ. In this lesson we use un and d. Cambridge also separates a progression (the list of terms) from a series (the sum of the terms). We study sums in Series #2.
5Arithmetic Sequences: Going Up or Down by the Same Amount
An arithmetic sequence has a constant common difference: each term is found by adding (or subtracting) the same number.
where a = first term, d = common difference, n = position of the term. Common difference: d = u2 − u1 = u3 − u2 = ...
Why (n − 1)? Think of a staircase
To go from the first term to the nth term, you take only (n − 1) steps. From u1 to u2 is 1 step, to u3 is 2 steps. So you add d exactly (n − 1) times, not n times.
| n | Term un | Using a and d |
|---|---|---|
| 1 | 2 | a |
| 2 | 2.5 | a + d |
| 3 | 3 | a + 2d |
| 4 | 3.5 | a + 3d |
| n | ... | a + (n − 1)d |
Link to linear functions (remember Grade 10)
un = a + (n − 1)d can be written as un = dn + (a − d). This is a linear function with gradient d. So the points (n, un) always lie on one straight line.
Left: an arithmetic sequence (straight line). Right: a geometric sequence (curve). The points exist only for whole numbers n. The dashed line just guides your eye.
Guided example 1: cloud layer width
Question: The 1st cloud layer is 2 cm wide, and each next layer is 0.5 cm wider. How wide is the 7th layer?
| Step | Work | Reason |
|---|---|---|
| 1 | a = 2; d = 0.5; n = 7 | Read the known values from the question. |
| 2 | un = a + (n − 1)d | The difference is constant, so it is arithmetic. |
| 3 | u7 = 2 + (7 − 1)(0.5) | Substitute. There are 6 steps from layer 1 to layer 7. |
| 4 | u7 = 2 + 3 = 5 cm | Calculate. |
| 5 | Check: 2, 2.5, 3, 3.5, 4, 4.5, 5 | The end of the list matches. Safe. |
In the Islamic boarding school tradition, istiqamah means being consistent. A hadith from Aisha (may God be pleased with her), reported by Bukhari and Muslim, means roughly: the deeds most loved by God are those done regularly, even if they are small.
Bridge to maths: a small habit that grows slowly but steadily is an arithmetic sequence. Say you study 10 minutes on day 1, then add 2 minutes each day. Then un = 10 + 2(n − 1). On day 15: u15 = 10 + 14 × 2 = 38 minutes.
This is a maths model to practise a concept. It is not a measure of the value of worship.
6Geometric Sequences: Multiplying by the Same Amount
A geometric sequence has a constant common ratio: each term is found by multiplying the previous term by the same number.
where a = first term, r = common ratio. Common ratio: r = u2 / u1 = u3 / u2 = ...
The form a · rn−1 is an exponential function (remember Grade 10, Series #13). That is why its graph curves upward quickly, as in the picture above.
| Ratio r | What the sequence does | Example |
|---|---|---|
| r > 1 | grows faster and faster | 2, 6, 18, 54, ... (r = 3) |
| 0 < r < 1 | shrinks, more and more slowly | 80, 40, 20, 10, ... (r = ½) |
| r < 0 | signs switch + and − | 3, −6, 12, −24, ... (r = −2) |
Guided example 2: a chain message
Question: Dina sends a message to 3 friends (round 1). Each new receiver forwards it to 3 people who have not received it yet. How many new receivers are there in round 5?
| Step | Work | Reason |
|---|---|---|
| 1 | Sequence: 3, 9, 27, ... | Round 1 has 3 people, round 2 has 3 × 3 = 9. |
| 2 | a = 3; r = 9/3 = 3 | The ratio is constant, so it is geometric. |
| 3 | u5 = 3 · 35−1 | Substitute n = 5. |
| 4 | u5 = 3 · 81 = 243 | Work out 3⁴ = 81 first. |
This verse gives a comparison: one grain grows seven ears, and each ear has one hundred grains. According to the Tafsir of the Indonesian Ministry of Religious Affairs, the result is 7 × 100 = 700 grains from one grain. The verse is about the multiplied reward for people who give in the way of God.
We only borrow the idea of multiplying as an example of a geometric sequence. Suppose (in an ideal model, with no pests and no limit on land) all the grains are planted again. The number of grains each season: 1, 700, 490,000, ... so a = 1, r = 700, and u4 = 7003 = 343,000,000.
7Arithmetic or Geometric? How to Decide
Follow this flow. Start with differences (easier), then try ratios.
If both the differences and the ratios are not constant, do not give up. There are other types, for example quadratic sequences (section 8).
| Feature | Arithmetic | Geometric |
|---|---|---|
| Change | add or subtract a constant | multiply by a constant (this includes a fixed percentage change) |
| Parameter | common difference d | common ratio r |
| nth term | un = a + (n − 1)d | un = a · rn−1 |
| Graph of points | on a straight line | on a curve (exponential) |
| Related function | linear | exponential |
| Cambridge name | arithmetic progression (AP) | geometric progression (GP) |
| Example in this lesson | cloud layer width; Islamic calendar | chain message; multiplying grains |
8Next Level: Quadratic Sequences
What about 1, 4, 9, 16, 25? The differences are 3, 5, 7, 9, which are not constant, and the ratios are not constant either. But try finding the difference of the differences.
Each new square adds an L-shaped group of dots (orange): 3, 5, 7, 9 dots. The difference between the additions is always 2.
The second differences are constant (2, 2, 2). This means the formula is quadratic: un = pn² + qn + s, where 2p = the second difference. This connects to quadratic functions from Grade 10.
First differences: 3, 5, 7, 9. Second differences: 2, 2, 2. So 2p = 2 and p = 1.
Subtract n² from each term: 2 − 1 = 1; 5 − 4 = 1; 10 − 9 = 1. What is left is always 1.
So un = n² + 1. Check n = 5: 25 + 1 = 26 ✓.
9Shortcut: Relating Any Two Terms
You are not always given the first term. Use the link between any two terms:
| Type | Relationship | Use |
|---|---|---|
| Arithmetic | un = um + (n − m)d | find d from two terms |
| Geometric | un = um · rn−m | find r from two terms |
Arithmetic. Given u3 = 11 and u8 = 26. The positions differ by 5 steps and the values differ by 15, so d = 15 ÷ 5 = 3. The first term is a = 11 − 2 × 3 = 5. Then u20 = 5 + 19 × 3 = 62.
Geometric. Given u2 = 6 and u5 = 162. Then r³ = 162 ÷ 6 = 27, so r = 3. The first term is a = 6 ÷ 3 = 2. Then u7 = 2 · 3⁶ = 2 × 729 = 1,458.
10Case Study: Panjang Jimat and the Islamic Calendar
Cirebon. On the night of 12 Rabiul Awal, the royal palaces of Cirebon, including Keraton Kasepuhan, hold the Panjang Jimat ceremony to remember the birthday of the Prophet Muhammad (peace be upon him). Because the event follows the Islamic (Hijri) date, its date in the Gregorian calendar changes every year.
Pesantren and NU. Islamic boarding schools and madrasahs linked to NU teach Ilmu Falak (hisab), the science that calculates the movement of the moon and sun for worship and the calendar. According to NU Online, an Islamic month lasts 29 or 30 days, and an Islamic year lasts 354 or 355 days. NU decides the start of a month by sighting the new moon (rukyatul hilal), guided by falak calculations.
Official data and the model
The date of 12 Rabiul Awal (a national holiday for the Prophet's birthday) in three years in a row, according to the Ministry of Religious Affairs calendar and the Joint Decree of 3 Ministers:
| n | 12 Rabiul Awal | Hijri year | un (days since 1 Jan 2024) | Difference |
|---|---|---|---|---|
| 1 | 16 Sep 2024 | 1446 H | 259 | – |
| 2 | 5 Sep 2025 | 1447 H | 613 | 354 |
| 3 | 25 Aug 2026 | 1448 H | 967 | 354 |
The difference is always 354 days, so this is an arithmetic sequence with a = 259 and d = 354:
Test the model: n = 3 → 259 + 354 × 2 = 967 ✓ (it matches 25 August 2026).
Prediction for n = 4: u4 = 259 + 3 × 354 = 1,321, which is about 14 August 2027.
Filled dots: official data. White circles: model predictions, not official announcements.
Interpreting and judging the model (IB style)
- Meaning of the difference 354: about one Islamic year. Because 365 − 354 ≈ 11, the Gregorian date of the Prophet's birthday moves about 10–12 days earlier each year.
- Limit of the model: an Islamic year can be 354 or 355 days, so the prediction can be 1 day off. The official decision is still made by calculation, moon sighting, and a government meeting (sidang isbat).
- Lesson: a sequence model is good for an estimate, but real data can differ a little. Checking your assumptions is part of doing maths.
As a note, for 2025 the Ministry of Religious Affairs and the NU Falakiyah Council both set 1 Rabiul Awal 1447 H on 25 August 2025, so the Prophet's birthday fell on 5 September 2025.
11One Question, Three Styles
In an exam, the same question can appear in a different “language”. Practise reading it.
| Style | Question | What it asks |
|---|---|---|
| National (Indonesia) | A student practises for 10 minutes on the first day and adds 2 minutes every day. Find the formula for the nth term and the practice time on day 15. | formula and value |
| Cambridge | A student practises for 10 minutes on day 1 and increases the time by 2 minutes each day. Find the time on day 15. | Find (calculate) |
| IB (applications) | The time is modelled by un = 10 + 2(n − 1). Interpret the value 2 in this context and state one limitation of the model. | Interpret and evaluate |
The answers are in the Answer Key. Notice that the IB style asks you to interpret and evaluate the model, not only to calculate.
12Watch Out for Traps!
- Using n instead of (n − 1). In un = a + (n − 1)d, the first term must give back a when n = 1.
- Thinking a three-term pattern is certain. The sequence 1, 2, 4, ... can continue with 8 (geometric) or 7 (the differences grow 1, 2, 3). A pattern with no stated rule is not unique.
- Thinking a negative ratio is wrong. 3, −6, 12, ... has r = −2 and is still geometric.
- Mixing up un and Sn. un is one term. Sn is the sum of the first n terms (Series #2).
- Thinking the Islamic year is always 354 days. It can be 355 days, so the model is only an estimate.
13Mini Dictionary
| English | Indonesian | Note |
|---|---|---|
| sequence / progression | barisan | an ordered list of terms |
| term | suku | one number in a sequence |
| common difference | beda | written d (or b in Indonesian books) |
| common ratio | rasio | written r |
| hisab | calculation (astronomy) | a word from Ilmu Falak |
| istiqamah | consistent, steady | a value used in pesantren |
| mega mendung | rain clouds | name of a Cirebon batik pattern |
14Your Turn!
Work on paper first, then check the Answer Key.
Question 1. A piece of Mega Mendung cloth has 10 layers. The 1st layer is 2 cm wide and each next layer is 0.5 cm wider. (a) Find the common difference. (b) How wide is the 7th layer? (c) Which layer is 6 cm wide?
Question 2. Chain message: round 1 has 3 new receivers, and each receiver forwards it to 3 new people. (a) Find the common ratio. (b) How many new receivers are there in round 6? (c) In which round does the number of new receivers first go above 1,000?
Digital literacy bonus: chain messages can spread very fast. Check the source before you forward.
1530-Second Cheat Sheet
Arithmetic: constant difference d. un = a + (n − 1)d. Graph: straight line (linear).
Geometric: constant ratio r. un = a · rn−1. Graph: curve (exponential).
Any two terms: un = um + (n − m)d and un = um · rn−m.
Quadratic: constant second difference → quadratic formula, 2p = second difference.
Remember: it is (n − 1) steps from the first term. A model is only an estimate, so check its assumptions.
16Closing Quiz
1. The sequence 7, 11, 15, 19, ... The 10th term is ...
- A. 39
- B. 43
- C. 47
- D. 40
2. The sequence 5, 10, 20, 40, ... The common ratio and the 7th term are, in order, ...
- A. r = 5 and u7 = 320
- B. r = 2 and u7 = 640
- C. r = 2 and u7 = 320
- D. r = 2 and u7 = 70
3. In the model un = 259 + 354(n − 1) for the Prophet's birthday, the difference 354 means ...
- A. the birthday is always exactly 354 days apart, with no exceptions
- B. the gap between birthdays is about one Islamic year, so the Gregorian date moves about 11 days earlier
- C. the Gregorian date moves 11 days later
- D. the difference has no meaning
3 out of 3? Great, go on to Series #2 (sums of sequences).
Less than 3? Read sections 5 and 6 again, then try once more.
17Quick Reflection
- Which part changed the way you see patterns around you (batik, calendar, or study habits)?
- What small habit could you raise slowly, with a constant difference, for one month?
Series #2: Series (sums). What happens when we add up the terms of a sequence? You will learn how to add 1 + 2 + ... + 100 in a few seconds.
Practice Sheets C1–C6 for Series #1 are prepared as a separate file.
🔑 Answer KeyDid you try everything? Then you may peek. (tap to open)
Warm-Up
| No. | Answer | Note |
|---|---|---|
| a | 17 | Constant difference +3 (arithmetic). |
| b | 48 | Multiply by 2 each time (geometric). |
| c | 25 | Square numbers: 1², 2², 3², 4², 5². |
| d | 5, 8, 11 | The difference is always 3, so a linear function gives an arithmetic sequence. |
One Question, Three Styles
un = 10 + 2(n − 1) = 2n + 8. Day 15: u15 = 10 + 14 × 2 = 38 minutes. IB style: the number 2 means practice time goes up by 2 minutes per day. Limit of the model: a constant increase is not realistic for a long time (practice time cannot grow forever).
Your Turn
| No. | Answer | Working |
|---|---|---|
| 1a | d = 0.5 | Each layer is 0.5 cm wider. |
| 1b | 5 cm | u7 = 2 + 6(0.5) = 5. |
| 1c | layer 9 | 2 + (n − 1)(0.5) = 6 → (n − 1) = 8 → n = 9. The cloth has 10 layers, so it fits. |
| 2a | r = 3 | Round 1: 3; round 2: 9; 9 ÷ 3 = 3. |
| 2b | 729 | un = 3 · 3n−1 = 3n. u6 = 3⁶ = 729. |
| 2c | round 7 | 3⁶ = 729 (less than 1,000), 3⁷ = 2,187 (more than 1,000). |
Closing Quiz
| No. | Key | Explanation of each option |
|---|---|---|
| 1 | B | a = 7, d = 4, u10 = 7 + 9 × 4 = 43. A (39) is u9: one step too few. C (47) uses 10 × 4, which forgets (n − 1). D (40) has no basis. |
| 2 | C | r = 10 ÷ 5 = 2. u7 = 5 · 2⁶ = 320. A has the wrong ratio. B uses 2⁷ (forgets n − 1). D adds instead of multiplying. |
| 3 | B | A difference of 354 days ≈ one Islamic year, so the Gregorian date moves back about 365 − 354 ≈ 11 days. A is wrong because an Islamic year can be 355 days. C has the wrong direction. D is wrong: the difference always has a meaning in context. |
📖References and Verification Status
| No. | Source | Used for | Type |
|---|---|---|---|
| 1 | BSKAP Decree No. 046/H/KR/2025 (Mathematics Learning Outcomes, Phase F) | Number strand outcomes: sequences and series | Official (read through education website summaries: secondary) |
| 2 | Cambridge International AS & A Level Mathematics 9709, 2026–2027 syllabus, Pure Mathematics 1 | The “Series” topic; the word progression | Official (check directly on the Cambridge website) |
| 3 | Save My Exams, 9709 Pure 1: Sequences and Series | Symbols un, a, d, r; AP and GP | Secondary |
| 4 | IB Diploma Programme, Mathematics: applications and interpretation (subject brief); syllabus summaries | Sequences and series topic | Official (details: secondary) |
| 5 | Joint Decree of 3 Ministers on National Holidays 2024, 2025, 2026 and the Ministry of Religious Affairs Hijri calendar; reported by Kompas.tv, CNN Indonesia, Katadata, Liputan6 | Dates of 12 Rabiul Awal 1446, 1447, 1448 H | Official (read through media: secondary) |
| 6 | NU Online: “Batas Ketinggian pada Qathiy dan Imkanur Rukyah dalam Kajian Falakiyah”; “Tahap-tahap Penentuan Awal Bulan Qamariah Perspektif NU” | Months of 29/30 days, years of 354/355 days; falak in NU pesantren and madrasah | Organisation publication |
| 7 | Eprints UIN Walisongo (studies of the Falakiyah Council and the book Sullam al-Nayyirain) | The role of falak in NU tradition | Academic study |
| 8 | NU Online (sermons and studies of the hadith on consistent deeds, narrated by Bukhari and Muslim) | The value of istiqamah | Organisation publication |
| 9 | quran.nu.or.id and Tafsir Kemenag, QS. Al-Baqarah [2]: 261 | The idea of multiplying 1 → 700 | Official translation and commentary |
| 10 | Proceedings UINSA (Panjang Jimat tradition); Fattah (2023), UIN Sunan Gunung Djati; Good News from Indonesia | Panjang Jimat in the Cirebon palaces on the night of 12 Rabiul Awal | Academic study and media |
| 11 | E-journal UMAHA (iconography study of Mega Mendung); Farhan (2023), UIN Sunan Gunung Djati; Good News from Indonesia | Features of Mega Mendung, colour gradation, seven layers (popular description) | Academic study and media |
Note: the numbers for cloud layer width (2 cm, +0.5 cm), the 10-minute study model, the chain message, and the multiplying grains are made-up models for practice, not field data.
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