Predicting What Comes Next: Exploring Sequences and Progressions

In an arithmetic progression each term is the last one plus a fixed number d. Find any term, or the sum of many terms, with two formulas.

Key ideas

• An AP has a first term a and a common difference d = second term − first term.
• d can be negative (a falling sequence) or a fraction.
• To check for an AP, the difference between every pair of neighbouring terms must be the same.
• The number of terms n must be a positive whole number; reject other solutions.
• Three numbers in an AP are easiest to write as a − d, a, a + d.

Remember

Watch out

• Using nd instead of (n − 1)d.
• Taking d as first − second instead of second − first.
• Keeping a negative or fractional value of n.

Worked example