KBAT: Progressions in Real Contexts

Q: When do I use the term formula and when the sum formula?

Use the term formula TnT_nTn when the question asks for the value at one particular step, one week, one month, one row. Use the sum formula SnS_nSn when it asks for a running total up to a point. Reading the question for 'the amount in week 5' versus 'the total after 6 weeks' decides it.

Q: How is Add Math Paper 2 marked on these questions?

Paper 2 is 2 hours 30 minutes and 100 marks, and marking is analytic, method marks are awarded line by line. Identifying the progression, choosing the correct formula and substituting correctly each earn credit, so show every stage.

Q: What is the most common mistake with progressions in context?

Confusing a single term with a total, answering with TnT_nTn when the question wanted SnS_nSn, or the reverse. The second common slip is mislabelling the sequence and using an arithmetic formula on a geometric situation.

What makes this a KBAT question

A routine progressions question tells you the first term and the common difference or ratio. A real-context KBAT question tells you a story and expects you to decide, first, which progression is hidden inside it, arithmetic when a fixed amount is added each step, geometric when a fixed factor multiplies each step, and then which formula answers the question: a single term $T_n$ , or a total $S_n$ .

That is higher-order thinking, Kemahiran Berfikir Aras Tinggi: the formulae are familiar, but nothing labels the sequence for you, and a comparison of two plans forces you to reason, not just compute. In Add Math this rewards students who read the structure of a situation before reaching for a formula.

One worked problem, in the style of Paper 2

This is an original question written in the style of SPM Paper 2. Try it before reading the solution.

Q1[8 marks]

A school runs a six-week tree-planting drive and considers two plans. Under Plan A the students plant $20$ trees in week 1 and $12$ more trees each week than the week before.

Under Plan B they plant $5$ trees in week 1 and double the number of trees each week. (a) Find the total number of trees planted under Plan A over the six weeks.

(b) Determine the first week in which the weekly number of trees under Plan B exceeds the weekly number under Plan A. (c) Decide which plan plants more trees in total over the six weeks.

Show worked solution

Understand. Plan A adds a fixed $12$ trees each week, so it is an arithmetic progression with first term $a=20$ and common difference $d=12$ .

Plan B multiplies by a fixed factor $2$ each week, so it is a geometric progression with first term $a=5$ and common ratio $r=2$ . Part (b) compares single weekly terms; parts (a) and (c) compare totals.

Plan. Use $S_n=\frac{n}{2}[2a+(n-1)d]$ for Plan A's total, the $n$ -th terms $T_n$ of each progression for the week-by-week comparison, and $S_n=\frac{a(r^{n}-1)}{r-1}$ for Plan B's total.

Execute and check. (a) Plan A total over $6$ weeks:

S_6 = \frac{6}{2}\,[\,2(20)+(6-1)(12)\,] = 3\,[\,40+60\,] = 3(100) = 300

So Plan A plants $300$ trees. (b) The weekly terms are $T_n(\text{A})=20+(n-1)(12)=12n+8$ and $T_n(\text{B})=5(2)^{\,n-1}$ .

Compare them week by week:

Week n	Plan A: $12n+8$	Plan B: $5\cdot 2^{n-1}$
1	20	5
2	32	10
3	44	20
4	56	40
5	68	80
6	80	160

Plan B first exceeds Plan A in week 5 ( $80>68$ ), having been smaller in weeks 1 to 4.

S_6 = \frac{5(2^{6}-1)}{2-1} = 5(64-1) = 5(63) = 315

Plan B plants $315$ trees against Plan A's $300$ . Since $315>300$ , Plan B plants more in total over the six weeks.

Check. Adding Plan B's weekly numbers directly gives $5+10+20+40+80+160=315$ , and Plan A gives $20+32+44+56+68+80=300$ , confirming both totals.

Finding a sensible first step

When a progressions question is dressed up as a story, the first step is never a formula, it is to identify the progression. Write out the first three or four terms from the words and look at how each term is built from the one before.

If a fixed amount is added, it is arithmetic, so find $a$ and $d$ . If a fixed factor multiplies, it is geometric, so find $a$ and $r$ .

The next decision is just as important: does the question ask for a single term (the amount in one week, one month, one row) or a running total (everything up to that point)? A single term needs $T_n$ ; a total needs $S_n$ .

Naming the progression and choosing between $T_n$ and $S_n$ turns almost any real-context question into standard Add Math.

What markers reward

Marking is analytic, so method marks are awarded line by line. On a real-context progressions question a marker looks for:

The progression correctly identified, arithmetic or geometric, with $a$ and either $d$ or $r$ stated.
The right choice between a term formula $T_n$ and a sum formula $S_n$ for what the question asks.
Correct substitution into $S_n=\frac{n}{2}[2a+(n-1)d]$ or $S_n=\frac{a(r^{n}-1)}{r-1}$ , with the working shown.
For an inequality like 'first week that exceeds', a clear comparison and a whole-number answer that is justified.
A decision stated in words for the 'which plan' part, supported by the totals.
Answers left as exact values where possible and interpreted back into the context.

How a teacher helps

The step students miss is the first one, deciding whether a story is arithmetic or geometric, so that is where our teachers spend time. In a one-to-one lesson we write out the opening terms together and ask 'added or multiplied?'

aloud, then choose between $T_n$ and $S_n$ before any substitution, so the habit holds under exam pressure. We also practise stating a decision in words, because comparison questions carry a reasoning mark.

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Frequently asked questions

How do I tell an arithmetic progression from a geometric one in a word problem?

Write the first few terms and see how each is made from the one before. If you add the same amount each time, it is arithmetic (find $a$ and $d$ ).

If you multiply by the same factor each time, it is geometric (find $a$ and $r$ ).

When do I use the term formula and when the sum formula?

Use the term formula $T_n$ when the question asks for the value at one particular step, one week, one month, one row. Use the sum formula $S_n$ when it asks for a running total up to a point.

Reading the question for 'the amount in week 5' versus 'the total after 6 weeks' decides it.

How is Add Math Paper 2 marked on these questions?

Paper 2 is 2 hours 30 minutes and 100 marks, and marking is analytic, method marks are awarded line by line. Identifying the progression, choosing the correct formula and substituting correctly each earn credit, so show every stage.

What is the most common mistake with progressions in context?

Confusing a single term with a total, answering with $T_n$ when the question wanted $S_n$ , or the reverse. The second common slip is mislabelling the sequence and using an arithmetic formula on a geometric situation.

Source:SRC-DSKP-EN SRC-FORMAT

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

KBAT: Progressions in Real-World Contexts