An arithmetic progression, or AP, is a sequence in which consecutive terms differ by a constant amount. This constant is called the common difference d. It may be positive, zero or negative. Always subtract the earlier term from the next term in the same order.
Find a term without listing everything
If the first term is a, then the nth term is aₙ = a + (n − 1)d. There are n − 1 jumps from the first term to the nth term. This explains why the formula contains n − 1 rather than n.
To test whether a number is a term of an AP, set a + (n − 1)d equal to that number and solve for n. A term position must be a positive integer. A fractional or non-positive result is not a valid position.
- For 9, 6, 3, 0, … the common difference is −3.
- A sequence of square numbers is not an AP because its differences are not constant.
Sum of the first n terms
The sum is Sₙ = n/2 × [2a + (n − 1)d]. If the last term l is known, use Sₙ = n(a + l)/2. Pairing the first and last terms explains why their average appears in the formula.
Distinguish a term from a sum. The tenth term is one number in the sequence; the sum of ten terms combines the first ten numbers. State which one the question asks for before calculating.
Put it into practice
A worked example
For the AP 5, 8, 11, … find the tenth term and the sum of ten terms.
- a = 5, d = 3 and n = 10.
- a₁₀ = 5 + 9 × 3 = 32.
- S₁₀ = 10(5 + 32)/2 = 185.
Answer: Tenth term: 32. Sum of the first ten terms: 185.
Try it yourself
Have a go before revealing the answer.
Find the twentieth term of 2, 6, 10, …
Show answer
2 + 19 × 4 = 78.
Answer: 2 + 19 × 4 = 78.
Is 31 a term of 4, 10, 16, …?
Show answer
No. 4 + (n − 1)6 = 31 gives n = 5.5, which is not a positive integer.
Answer: No. 4 + (n − 1)6 = 31 gives n = 5.5, which is not a positive integer.
Watch out for these
- Do not substitute n in place of n − 1 in the nth-term formula.
- Keep the sign of d when an AP is decreasing.
Continue with official resources
Match these concept notes to your current textbook and syllabus. They are independent explanations, not official board publications.