Working with arithmetic progressions

An arithmetic progression, or AP, is a sequence in which every term is found by adding the same fixed number to the term before it. That fixed number is the common difference $d$.

In Add Math you use this method to describe any situation that grows or shrinks in equal steps, a savings plan that adds the same amount each month, seats that increase by a fixed number in each row of a hall, or a stack whose height rises evenly.

Two workhorse formulas let you jump straight to an answer without listing every term. The first gives the value of any single term; the second gives the running total of the first $n$ terms.

Together they answer almost every AP question you will meet.

Reach for this method whenever a list of numbers goes up or down by the same amount each time. Check by subtracting neighbouring terms: if $T_2 - T_1 = T_3 - T_2$, the sequence is arithmetic and that shared value is $d$.

Exam questions signal an AP with phrases such as 'increases by a constant amount', 'equal instalments', or a table of values with a steady gap.

If instead the terms are multiplied by a fixed ratio at each step, that is a geometric progression and needs different formulas. Watch closely for the word sum, it tells you to use $S_n$ rather than $T_n$.

Deciding early which formula the question wants saves time and protects your working marks.

1. 1

   Identify a and d

   Write down $a$ (the first term) and find $d$ by subtracting any term from the one after it: $d = T_2 - T_1$.

2. 2

   Choose the right formula

   Use $T_n = a + (n-1)d$ if the question asks for a single term; use $S_n = \frac{n}{2}[2a+(n-1)d]$ if it asks for a sum.

3. 3

   Set up equations if a or d is unknown

   If two terms are given, write each as $a + (n-1)d$, then solve the pair of simultaneous equations for $a$ and $d$.

4. 4

   Substitute carefully

   Put the known numbers in, keeping $(n-1)$, not $n$, inside the bracket.

5. 5

   Simplify to a clean value

   Work through the arithmetic one line at a time and state the final answer with its units, if any.

6. 6

   Check

   Verify by listing a few terms, or by testing your $a$ and $d$ in one of the terms the question gave you.

• Using $n$ instead of $(n-1)$ in the term formula, the first term needs zero steps of $d$, so the multiplier is $(n-1)$.
• Mixing up $T_n$ and $S_n$: a 'term' question wants a single value, a 'sum' question wants a total.
• Getting the sign of $d$ wrong in a decreasing sequence, where $d$ is negative.
• Solving the two-term equations by guessing instead of subtracting to eliminate $a$ cleanly.
• Forgetting to substitute the found $d$ back to get $a$, and stopping half way.

Most AP mistakes come down to one slip, a student writes $n$ where the formula needs $(n-1)$, or reaches for $T_n$ when the question quietly asks for a sum. In one-to-one lessons our teachers watch each line you write and stop at the exact moment the error appears, so the wrong habit never sets in.

We also train you to show the substitution line clearly, so your working is easy to follow and easy to credit line by line. Every teacher on spmaddmath.com.my is experienced.

Lessons are online and taught in English, matched to your own pace.