Arithmetic Series (OCR A Level Maths A: Pure): Revision Note

Arithmetic series

How do I find the sum of an arithmetic series?

• An arithmetic series is the sum of the terms of an arithmetic sequence

• The following formulae will let you find the sum of the first n terms of an arithmetic series:

  $S_n=\frac{n}{2}(2a+(n-1\left)d\right)$   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 sequences

How do I prove the arithmetic series formula?

• Learn this proof of the arithmetic series 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