Progressions, SPM Additional Mathematics Form 4

A progression is a sequence of numbers that follows a fixed rule from one term to the next. In SPM Additional Mathematics this is Chapter 5 of Form 4, and it lives in the Algebra learning area.

The chapter takes the everyday idea of a number pattern, 3, 7, 11, 15, … or 2, 6, 18, 54, …, and gives you exact tools to describe it, extend it as far as you like, and add up as many terms as you need.

The syllabus splits Progressions into two content standards. The first, arithmetic progressions (AP), deals with sequences that climb or fall by the same fixed amount each step, the common difference $d$.

The second, geometric progressions (GP), deals with sequences where each term is the one before it multiplied by the same fixed number, the common ratio $r$. Almost every question in the chapter comes down to identifying $a$, the first term, together with $d$ or $r$, and then choosing the right formula.

Add Math is an elective subject, and Progressions is one of the more approachable chapters for building early confidence. Four of its main formulae are printed on the SPM formulae page, the arithmetic stays light, and the reasoning is usually short and direct.

That makes it an ideal place to practise the habit that earns method marks throughout the whole paper, writing every substitution clearly on its own line before reaching for the calculator.

By the end of the chapter you should be able to decide whether a given sequence is arithmetic, geometric, or neither; find any term without listing all the terms before it; add a long run of terms in a single line of working; and, for a geometric progression whose terms keep shrinking, find the single value that the endless sum settles down to. These skills reappear in later topics and in real problems about savings, loan repayments and steady growth or decay.

Progressions is organised into two content standards. These codes and titles come directly from the KSSM DSKP, knowing them tells you exactly what a question is testing.

Both standards share the same shape, identify the first term and the constant step, then apply a formula, but they behave very differently. An AP grows in a straight line, while a GP grows (or shrinks) by repeated multiplication, which is why a GP can climb steeply or fade towards a limit.

Being fluent with both lets you handle almost any sequence the paper puts in front of you.

Sequence, term and progression. A sequence is an ordered list of numbers, and each number is a term, written $T_{1}, T_{2}, T_{3}, \dots$.

A progression is a sequence built by a fixed rule. The two rules you study are adding a constant (arithmetic) and multiplying by a constant (geometric).

Recognising an arithmetic progression. A sequence is an AP when the difference between each term and the one before it is constant.

Test it by computing $T_{2}-T_{1}$, $T_{3}-T_{2}$ and $T_{4}-T_{3}$; if they are all equal, that shared value is the common difference $d$, and you may quote it as your justification.

The nth term of an AP. Once you know the first term $a$ and the common difference $d$, any term follows without listing the ones before it.

This is the first formula you should reach for whenever a question asks for a particular term.

$$T_{n}=a+(n-1)d$$

The sum of the first n terms of an AP. To add the first $n$ terms, use $S_{n}$.

When you happen to know the last term $l=T_{n}$, the equivalent form $S_{n}=\frac{n}{2}(a+l)$ is often quicker to use.

$$S_{n}=\frac{n}{2}\big(2a+(n-1)d\big)$$

Recognising a geometric progression. A sequence is a GP when the ratio of each term to the one before it is constant.

Test it by computing $\frac{T_{2}}{T_{1}}$, $\frac{T_{3}}{T_{2}}$ and $\frac{T_{4}}{T_{3}}$; if they are all equal, that shared value is the common ratio $r$. Note that $r$ can be a fraction (terms shrink) or negative (terms alternate in sign).

The nth term of a GP. With the first term $a$ and the common ratio $r$, any term is a power of $r$ away from the start.

The index is $n-1$, so the first term corresponds to $r^{0}=1$.

$$T_{n}=ar^{\,n-1}$$

The sum of the first n terms of a GP. Two equivalent versions are printed; use the one that keeps the signs tidy.

When $r>1$ the first form is neater, and when $r<1$ the second avoids negative numbers on the top and bottom.

$$S_{n}=\frac{a(r^{n}-1)}{r-1}=\frac{a(1-r^{n})}{1-r},\ r\neq 1$$

The sum to infinity of a GP. When the terms keep shrinking, that is, when $|r|<1$, the running total gets closer and closer to a single finite value.

This is the sum to infinity. It applies only when $|r|<1$; if $|r|\ge 1$ the terms do not shrink and there is no finite sum.

$$S_{\infty}=\frac{a}{1-r},\quad |r|<1$$

Finding $a$ and $d$ (or $r$) from two given terms. Many questions give you two terms, such as $T_{3}=11$ and $T_{7}=27$, and ask for the progression.

Write each as an equation in $a$ and $d$ (or $a$ and $r$) using the nth-term formula, then solve the pair simultaneously. For the AP example, $a+2d=11$ and $a+6d=27$ give $4d=16$, so $d=4$ and $a=3$.

Modelling real situations. A fixed yearly salary increment, seats increasing by a constant number per row, or equal monthly savings all form arithmetic progressions.

Repeated percentage growth, depreciation of a vehicle's value, or a bouncing ball losing a fixed fraction of its height each bounce form geometric progressions, and a recurring decimal such as $0.4444\ldots$ is exactly the sum to infinity of a GP. Reading a word problem well means first deciding which model fits.

Progressions can be tested in either written paper. Paper 1 (3472/1) lasts 2 hours and carries 80 marks: Section A has 12 questions worth 64 marks that you answer all of, and Section B has 3 questions worth 16 marks from which you answer 2.

Paper 2 (3472/2) lasts 2 hours 30 minutes and carries 100 marks across Section A (7 questions, 50 marks, answer all), Section B (4 questions, 30 marks, answer 3) and Section C (4 questions, 20 marks, answer 2).

Across both papers the items are limited-response subjective and structured questions, they are marked using analytic scoring, and you sit them with a non-programmable scientific calculator. Questions are set across a range of difficulty, the format uses a low-to-high difficulty ratio of 5 : 3 : 2, so you can expect both direct substitution items and longer, multi-step word problems.

We do not predict how many marks any single chapter will carry, because that varies from year to year.

This chapter suits the structured, show-your-working format well. A typical short item asks you to find a specific term or a sum; a longer item might give you two terms, ask you to find $a$ and $d$, and then use them to answer a real-context question.

Because the scoring is analytic, each correct substitution, each solved value, and each final answer can earn its own mark, so writing the formula, then the substitution, then the result protects your marks even if a single arithmetic step slips.

Most marks lost in this chapter come from a handful of avoidable habits. Read these before every practice set until they become second nature.

• Confusing $d$ and $r$. Before choosing a formula, test the sequence: a constant difference means arithmetic, a constant ratio means geometric. Applying an AP formula to a GP, or the other way round, is the single most common error in this chapter.
• Slipping on the $n-1$ index. The nth term uses $n-1$, not $n$: $T_{n}=a+(n-1)d$ and $T_{n}=ar^{n-1}$. Using $n$ by mistake shifts every answer by one term.
• Using the sum to infinity when it does not apply. $S_{\infty}=\frac{a}{1-r}$ is valid only when $|r|<1$. If $|r|\ge 1$ the terms do not shrink, so there is no finite sum, check $r$ first.
• Assuming every formula is given. The four AP and GP formulae for $T_{n}$ and $S_{n}$ are on the formulae page, but the sum to infinity is not. Memorise $S_{\infty}=\frac{a}{1-r}$ so it is ready in the exam.
• Miscounting the number of terms. When a question asks how many terms lie between two positions, count inclusively and check with the nth-term formula. For example, the terms from $T_{5}$ to $T_{20}$ number $20-5+1=16$, not 15.
• Forgetting that $n$ must be a positive integer. When solving 'which term first exceeds …', you often get a non-whole value of $n$; round up to the next whole number, because a term only exists at a whole-number position.

None of these mistakes is about ability, each one is a habit, and habits are fixable. Test for AP or GP before you start, write the index carefully, and check the condition on $r$; do that and your accuracy in this chapter climbs quickly.

Progressions rewards a steady, ordered approach. Work through these steps in turn, then use the resources below to revise and test yourself.

Aim for short, frequent sessions rather than one long cram. Do a couple of AP items and a couple of GP items each day, always writing the formula before the numbers.

When a step feels solid, move on; when it does not, slow down and repeat it. Spread over a week or two, this quiet practice turns Progressions into some of your most dependable marks.

1. 1

   Learn the five formulae

   Memorise which four are given ($T_{n}$ and $S_{n}$ for AP and GP) and which one is not, the sum to infinity, and note the condition $|r|<1$.

2. 2

   Drill identification

   Practise deciding whether a sequence is arithmetic, geometric, or neither by testing the difference and the ratio, and justify your answer each time.

3. 3

   Find a and d or r

   Given two terms, form two equations from the nth-term formula and solve them simultaneously to pin down the progression.

4. 4

   Work full word problems

   Translate salary, savings, growth and recurring-decimal problems into an AP or GP, then answer them, writing every substitution.

5. 5

   Time yourself and review

   Attempt mixed practice under a clock, then re-read your common-mistake list before the next session.

• Progressions, Revision Notes
• Progressions, Common Mistakes
• Progressions, Practice Questions
• Progressions, Paper 2 Answering Guide
• Progressions, Key Terms
• Progressions, Worked Examples (Easy)
• Progressions, Worked Examples (Medium)
• Progressions, Worked Examples (KBAT)