Progressions, Glossary

Knowing the vocabulary of Progressions precisely is worth easy marks in SPM Add Math. Questions often turn on a single keyword, read it correctly and the method follows; misread it and even good algebra earns nothing.

Because the SPM paper is bilingual (BM/EN), each term below shows the English and Malay wording, so your child recognises it whichever way a question is phrased.

• Sequence, A sequence is an ordered list of numbers, called terms, that follow a rule, such as $2, 5, 8, 11, \dots$. Each term has a position, and the general term $T_n$ gives the value at position $n$. Recognising the pattern lets us predict any later term. →
• Arithmetic Progression, An arithmetic progression is a sequence in which each term is found by adding a fixed number, the common difference $d$, to the previous term. Its $n$th term is $T_n=a+(n-1)d$, where $a$ is the first term. Examples include $3, 7, 11, 15,\dots$ with $d=4$. →
• Common Difference, The common difference $d$ is the fixed amount added between consecutive terms of an arithmetic progression, found by $d=T_n-T_{n-1}$. A positive $d$ makes the sequence increase, a negative $d$ makes it decrease. It stays the same between every pair of neighbouring terms. →
• Geometric Progression, A geometric progression is a sequence in which each term is found by multiplying the previous term by a fixed number, the common ratio $r$. Its $n$th term is $T_n=ar^{n-1}$, where $a$ is the first term. For example $2, 6, 18, 54,\dots$ has $r=3$. →
• Common Ratio, The common ratio $r$ is the fixed factor by which each term of a geometric progression is multiplied to get the next, found by $r=\frac{T_n}{T_{n-1}}$. If $|r|>1$ the terms grow, and if $|r|<1$ they shrink towards zero, which allows a sum to infinity. →
• nth Term, The $n$th term, written $T_n$, is a formula giving the value at any position $n$ in a progression without listing all earlier terms. For an arithmetic progression $T_n=a+(n-1)d$, and for a geometric one $T_n=ar^{n-1}$. Substituting a position number returns that term directly. →
• Sum of the First n Terms, The sum of the first $n$ terms, written $S_n$, adds up a progression from the first term to the $n$th. For an arithmetic progression $S_n=\frac{n}{2}[2a+(n-1)d]$; for a geometric one $S_n=\frac{a(r^{n}-1)}{r-1}$. Any single term can be recovered as $T_n=S_n-S_{n-1}$. →
• Sum to Infinity, The sum to infinity, $S_\infty$, is the total of a geometric progression that never ends but settles on a finite value. It exists only when $|r|<1$, and is given by $S_\infty=\frac{a}{1-r}$. As terms shrink towards zero, the running total approaches this limit. →

Do not just memorise definitions, practise using each term inside a full solution, the way a marker expects to see it. In the Progressions chapter especially, stating the right definition or condition in your working can itself earn a method mark.

Our teachers check that a student can both recall a term and deploy it fluently under exam conditions.

Vocabulary sticks best when it is tied to doing rather than reading. Instead of learning the Progressions terms as a flat list, meet each one inside a worked question and say the step aloud in words as you write it, a term you can use in a sentence is a term you understand.

It also helps to link related terms together rather than in isolation, since Progressions questions usually combine several ideas at once. Because the SPM paper is bilingual, glance once at the Malay and English wording of each term side by side, so the language a question is set in never throws you.

If a term keeps feeling slippery, that is usually a sign the underlying idea needs another look, not the word itself, and that is exactly the kind of gap a one-to-one lesson closes quickly.