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Arithmetic Series

Maths: Edexcel A Level Pure Maths, topic 4.2.7

Arithmetic Series

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

Series

Series

  • The sum of the terms of a sequence is called a series.
  • For the arithmetic sequence with a1=5a_1 = 5 and d=2d = 2, the first four terms are:
  • 5,7,9,115,7,9,11
  • The series is then given by:
  • 5+7+9+11=325 +7+9+11 = 32
General formula

General formula

  • We can write the sum of the first nn terms of an arithmetic series with a difference dd as:
  • Sn=a1+(a1+d)+...+(a1+(n−2)d)+(a1+(n−1)d)S_n = a_1 + (a_1+d)+ ...+(a_1+(n-2)d)+(a_1+(n-1)d)
  • Reversing the order of the terms, we can also write the sum as
  • Sn=(a1+(n−1)d)+(a1+(n−2)d)+...+(a1+d)+a1S_n = (a_1+(n-1)d)+(a_1+(n-2)d)+...+(a_1+d)+a_1
General formula

General formula

  • Adding the two forms of SnS_n, we can see that the terms match up:
  • 2Sn=(2a1+(n−1)d)+(2a1+(n−1)d)+...+(2a1+(n−1)d)+(2a1+(n−1)d)2S_n = (2a_1+(n-1)d) + (2a_1+(n-1)d) +...+(2a_1+(n-1)d)+(2a_1+(n-1)d)
  • Because there are nn terms in the series, we can simplify this sum to:
  • 2Sn=n(2a1+(n−1)d)→Sn=12n(2a1+(n−1)d)2S_n=n(2a_1+(n-1)d) \rightarrow S_n = \frac{1}{2}n(2a_1+(n-1)d)
Sum of natural numbers

Sum of natural numbers

  • If we set a1=1a_1=1 and d=1d=1, we can use the equation for the sum of an arithmetic series to find the sum of the first nn natural numbers:
  • Sn=12n(2+(n−1))=n(n+1)2S_n = \frac{1}{2}n(2 + (n-1)) = \frac{\small n(n+1)}{\small 2}

Test yourself on Arithmetic Series

  • What is the equation for the sum of the first nn natural numbers?

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