Arithmetic Progressions

Lists that go up or down by the same amount each time: find any term, and the sum of the first n terms.

Key ideas

• In an AP, the common difference d = (any term) − (the term before it) is the same all the way along.
• d can be positive, negative or zero.
• The nth term counts n − 1 steps of d from the first term a.
• The sum of the first n terms is the number of terms times the average of the first and last terms.
• Three numbers a, b, c are in AP exactly when b − a = c − b, that is 2b = a + c.
• The nth term can be found from the sums: a_n = S_n − S_n−1.

Remember

Watch out

• Using n instead of n − 1 in the nth-term formula.
• Mixing up a_n (one term) and S_n (a total).
• Losing the minus sign when d is negative.

Worked example