12.4 AP and GP Formulas

Part 1: Arithmetic Progressions (AP)

What is an Arithmetic Progression?

A sequence is simply a list of numbers written in a specific order, following a rule.

An arithmetic progression (AP) is a special type of sequence where you always add the same number to get from one term to the next.

That fixed number you add each time is called the common difference, written as d.

Example: The sequence 5, 8, 11, 14, 17, ... is an AP.

• From 5 to 8: add 3
• From 8 to 11: add 3
• From 11 to 14: add 3

So the common difference is d = 3.

The Key Variables for an AP

The Terms of an AP — Spotting the Pattern

If you write out the first few terms, each term is built from the first term a plus multiples of d:

Notice: the nth term has (n − 1) lots of d added to a.

Formula 1: The nth Term of an AP

$\boxed{T_n = a + (n-1)d}$

• Tₙ = the value of the nth term
• a = first term
• n = position of the term
• d = common difference

This formula lets you find any term in the sequence without listing every term before it.

Worked Example 1 — Finding the number of terms

Question: How many terms are in the AP: −17, −14, −11, −8, ..., 58?

Step 1: Identify the values.

• a = −17 (first term)
• d = −14 − (−17) = 3 (common difference)
• Last term = 58

Step 2: Set the nth term formula equal to 58.

$58 = -17 + (n-1)(3)$

Step 3: Solve for n. $58 + 17 = 3(n-1)$ $75 = 3(n-1)$ $n - 1 = 25$ $n = 26$

Answer: There are 26 terms in this AP.