What is an arithmetic progression?
An arithmetic progression (AP) is a sequence in which each term differs from the previous one by a fixed number called the common difference d. For example 3, 7, 11, 15, … has first term a = 3 and d = 4.
Key formula
nth term: Tn = a + (n − 1)d
Sum of n terms: Sn = n/2 [2a + (n − 1)d]
or Sn = n/2 (a + l), where l is the last term.
Finding a term and the common difference
The common difference is any term minus the one before it: d = T2 − T1. Once you know a and d, the nth-term formula gives any term directly.
- For 5, 9, 13, …: d = 4, so T7 = 5 + 6(4) = 29.
- To find which term equals a value, solve Tn for n.
Worked example
An AP has a = 1 and d = 2. Find T10 and S10.
T10 = 1 + (10 − 1)(2) = 1 + 18 = 19.
S10 = 10/2 [2(1) + 9(2)] = 5 × 20 = 100.
Sum of terms
Use Sn = n/2 (a + l) when the first and last terms are known. For 10 + … + 100 with 10 terms, S = 10/2 (10 + 100) = 5 × 110 = 550.
Remember
- d can be negative — the sequence then decreases.
- The (n − 1) in Tn is easy to forget; count the steps, not the terms.
- Check whether the question asks for a term (Tn) or a sum (Sn).