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Sigma Notation

Maths: Edexcel A Level Pure Maths, topic 4.2.6

Sigma Notation

The Greek letter Sigma Σ is used to represent the sum of a collection of terms.

Sigma

Sigma

  • Sigma notation, Σ, is a shorthand notation for sums in which each term can be written as a general expression.
  • We write an integer variable rr below Σ and give it a starting value.
  • The highest value of rr is written above the Σ.
  • The general expression is written to the right of Σ which is usually a function of rr.
  • The highest and lowest values of rr are called the limits of the sum.
Evaluating sums

Evaluating sums

  • To evaluate the sum, we work out the first term by substituting the initial value of rr into the general expression.
  • The next term is found by adding 1 to the initial value of rr, and then substituting it in to the general expression.
  • The value of rr increases by 1 for each term, and the sum ends when rr is equal to the highest value.
Example

Example

  • Evaluate the sum ∑r=244r2\sum_{r=2}^{4}4r^2.
Limits

Limits

  • The limits of the sum show that we have to add the terms up for r=r = 2, 3 and 4.
First term

First term

  • Substituting r=2r=2 into 4r24r^2 gives the first term:
  • 4r2=4×4=164r^2 = 4\times 4 = 16
Second term

Second term

  • Substituting r=3r=3 into 4r24r^2 gives the second term:
  • 4r2=4×9=364r^2 = 4\times 9 = 36
Final term

Final term

  • Substituting r=4r=4 into 4r24r^2 gives the third term:
  • 4r2=4×16=644r^2 = 4\times 16= 64
  • This is the final term, as r=4r=4 is the upper limit of the sum.
Find the sum

Find the sum

  • Adding each term together gives the value of the sum:
  • ∑r=24=16+36+64=116\sum_{r=2}^{4} = 16+36+64 = 116
Sum of one

Sum of one

  • An important sum you need to remember is ∑r=1n1=n\sum_{r = 1}^{n} 1=n.
  • We can rewrite this as 1+1+...+1+11+1+...+1+1 where there are nn terms, each equal to 1.
  • This means the sum is adding the number 1 to itself nn times, which is equal to 1×n=n1 \times n = n.

Test yourself on Sigma Notation

  • What is the sigma notation form of the following sum?
    10+17+26+3710+17+26+37

  • What is the sigma notation form of the following sum?
    3+5+7+93+5+7+9

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