The sigma notation
Edexcel International GCSE Further Pure MathematicsΒ· Section S5, Specification 5AΒ· 15 min read
1. What is Sigma Notation?β βββββ± 4 min
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Sigma Notation
A compact way to write the sum of terms f(k) where k takes every integer value from the lower limit a to the upper limit b, inclusive.
Example:
Sum of first 5 positive integers: (\sum_{k=1}^{5} k = 1+2+3+4+5 = 15)
The Greek capital letter Ξ£ (sigma) stands for 'sum of'. The variable below the sigma is called the index of summation, and can be any letter (k, n, r are most common). You substitute every integer value of the index from the lower limit up to the upper limit into the expression next to the sigma, then add all resulting terms together.
Identify the terms and calculate the value of (\sum_{k=2}^{4} (2k + 1))
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Step 1: Identify the limits: lower limit k=2, upper limit k=4. Substitute k=2, 3, 4 into the expression 2k+1:
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Step 2: Add the terms together:
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Final result: 21
Exam tip:
Always double-check you have included both the lower and upper limit values when expanding the sum - missing one of these is a common easy mark to lose.
2. Writing Sums Using Sigma Notationβ β ββββ± 5 min
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To rewrite an expanded sum in sigma notation, first identify the pattern of the terms, define your index variable and limits, then write a general expression for the nth term of the sequence.
Rewrite the sum (3 + 7 + 11 + 15 + 19 + 23) using sigma notation.
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Step 1: Identify the term pattern: each term increases by 4, so it is a linear sequence with nth term 4n - 1 (check: n=1 gives 3, n=2 gives 7, etc.)
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Step 2: Identify limits: there are 6 terms, so the index runs from n=1 to n=6
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Step 3: Write the sigma expression:
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Alternative valid form: you could use a different index, e.g. (\sum_{k=0}^{5} (4k + 3)), as long as the expanded terms match exactly.
Which of the following is a correct sigma notation for the sum 1 + 4 + 9 + 16 + 25?
A. (\sum_{k=1}^{5} 2k)
B. (\sum_{k=1}^{5} k^2)
C. (\sum_{k=0}^{4} k^2)
Reveal answer
B βOption B gives 1Β² + 2Β² + 3Β² + 4Β² +5Β² = 1+4+9+16+25, which matches the given sum. Option A gives even numbers, option C starts at k=0 so its first term is 0.
Exam tip:
If you are asked to write a sum in sigma notation, you can always check your answer by expanding the first and last terms to confirm they match the given sum.
3. Evaluating Algebraic Sigma Sumsβ β β βββ± 6 min
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You may be asked to evaluate sums with algebraic terms, or sums where the expression under sigma is a constant. For constant terms, you are just adding the constant as many times as the number of terms between the limits (inclusive).
Evaluate (\sum_{r=1}^{10} 3) and (\sum_{x=3}^{7} (x^2 + 2x))
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First sum: (\sum_{r=1}^{10} 3)
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Step 1: Number of terms = 10 - 1 + 1 = 10 terms, each equal to 3
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Second sum: (\sum_{x=3}^{7} (x^2 + 2x))
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Step 1: Substitute x=3,4,5,6,7:
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Step 2: Add all terms: 15 +24 +35 +48 +63 = 185
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Final results: 30 and 185 respectively
4. Common Pitfalls
Wrong move:
Only summing terms from lower limit +1 up to upper limit, missing the lower limit term
Why:
Students often forget to substitute the lower limit value, leading to an incorrect sum with one term missing
Correct move:
Always confirm you have (upper limit - lower limit + 1) terms in your expanded sum before adding them.
Wrong move:
Assuming the index always starts at 1
Why:
Limits can start at any integer, including negative numbers or zero, so substituting starting at 1 will give wrong terms
Correct move:
Always check the lower limit of the summation before substituting values for the index.
Wrong move:
Treating the index variable as a global variable
Why:
The index only exists within the sigma expression, so you cannot combine it with variables outside the sum
Correct move:
If the sum is part of a larger expression, treat the entire sigma sum as a single constant/variable value when simplifying the rest of the expression.
Wrong move:
Forgetting to apply all parts of the expression under sigma to each index value
Why:
For example, for (\sum (k + 2)), students sometimes only sum k and then add 2 once at the end, instead of adding 2 to every term
Correct move:
Substitute the index into the full expression each time, or expand the sum first if appropriate to separate constant and variable terms.
Wrong move:
Using non-integer values for the index
Why:
At IGCSE Further Pure level, the index of summation always takes consecutive integer values only
Correct move:
Only substitute integer values of the index from lower to upper limit, skipping no integers in between.
5. Quick Reference Cheatsheet
Task | Step-by-Step Method | Example |
|---|---|---|
Interpret sigma notation |
| (\sum_{k=1}^3 2k = 2(1)+2(2)+2(3) = 12) |
Write sum in sigma notation |
| Sum 2 +4 +6 +8 = (\sum_{n=1}^4 2n) |
Evaluate constant sum |
| (\sum_{r=2}^5 4 = (5-2+1) \times 4 = 16) |
What's Next
Now that you have mastered sigma notation, you are ready to apply it to arithmetic and geometric series, the next topic in the Edexcel IGCSE Further Pure Math series unit. Sigma notation will also appear later in your course when you study binomial expansions and other summation problems, so it is a critical foundational skill. Practice writing and evaluating sigma sums regularly to avoid simple errors in exam questions, and always check your expanded terms match the expected pattern before calculating the final sum.