Sum of Arithmetic Progressions (Cambridge International A-Level Mathematics)

Arithmetic Series

Sum of arithmetic progressions

How do I find the sum of an arithmetic progression?

• The sum of the terms of an arithmetic progression is sometimes called an arithmetic series

• The following formulae will let you find the sum of the first n terms of an arithmetic progression: $S_{n}=\frac{n}{2}(2a+(n-1)d)$ or $S_{n}=\frac{n}{2}(a+l)$

  • a is the first term
  • d is the common difference
  • l is the last term

• You can use whichever formula is more convenient for a given question

• The a and the d in those formulae are exactly the same as the ones used with arithmetic progressions

How do I derive the formula for the sum of an arithmetic progression?

• Learn this proof of the sum of an arithmetic progression formula – you can be asked to give it on the exam:

  • Write the terms out once in order

  • Write the terms out again in reverse order

  • Add the two sums together

    • The terms will pair up to give the same sum $2a+(n-1)d$
    • There will be n of these terms

  • Divide by two as two of the sums have been added together

Exam Hint The arithmetic series formulae are in the formulae booklet – you don't need to memorise them.

Worked Example