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Summation — Definition, Formula & Examples

Summation is the operation of adding a list of numbers. Summation notation uses the Greek capital letter sigma, Σ\Sigma, to write a long sum in one short expression.

For integers m≤nm \le n and terms aka_k defined for each integer kk with m≤k≤nm \le k \le n, the expression ∑k=mnak\sum_{k=m}^{n} a_k denotes am+am+1+⋯+ana_m + a_{m+1} + \cdots + a_n. The variable kk is the index of summation, mm is the lower limit, and nn is the upper limit. The sum contains n−m+1n - m + 1 terms.

Key Formula

∑k=mnak=am+am+1+am+2+⋯+an\sum_{k=m}^{n} a_k = a_m + a_{m+1} + a_{m+2} + \cdots + a_n
Where:
  • Σ\Sigma = Sigma, the symbol meaning add the terms
  • kk = Index of summation, which takes each integer value from m to n
  • mm = Lower limit, the first value of the index
  • nn = Upper limit, the last value of the index
  • aka_k = The general term, a rule giving each term from k

How It Works

Start the index at the lower limit and substitute it into the expression after the Σ\Sigma. Increase the index by 1 each time and substitute again. Stop after you substitute the upper limit, then add every result. The index letter does not affect the value, so ∑k=14k\sum_{k=1}^{4} k and ∑i=14i\sum_{i=1}^{4} i are equal.

Worked Example

Problem: Evaluate ∑k=15(2k+1)\sum_{k=1}^{5} (2k + 1).
Step 1: Substitute each value of kk from 1 to 5 into 2k+12k + 1. The sum has 5−1+1=55 - 1 + 1 = 5 terms.
(2⋅1+1)+(2⋅2+1)+(2⋅3+1)+(2⋅4+1)+(2⋅5+1)=3+5+7+9+11(2\cdot1+1) + (2\cdot2+1) + (2\cdot3+1) + (2\cdot4+1) + (2\cdot5+1) = 3 + 5 + 7 + 9 + 11
Step 2: Add the terms.
3+5+7+9+11=353 + 5 + 7 + 9 + 11 = 35
Step 3: Check with the properties of summation. Split the sum and pull out the constant factor. Note that adding 1 five times gives 5.
∑k=15(2k+1)=2∑k=15k+∑k=151=2(15)+5=35\sum_{k=1}^{5} (2k+1) = 2\sum_{k=1}^{5} k + \sum_{k=1}^{5} 1 = 2(15) + 5 = 35
Answer: ∑k=15(2k+1)=35\sum_{k=1}^{5} (2k + 1) = 35

Why It Matters

In statistics, the mean of a data set is written xˉ=1n∑i=1nxi\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i, and variance and standard deviation use the same notation. In calculus, the definite integral is defined as a limit of Riemann sums, so you need to read Σ\Sigma fluently before studying integration.

Common Mistakes

Mistake: Treating a constant term as if it appears only once, for example writing ∑k=153=3\sum_{k=1}^{5} 3 = 3.
Correction: The constant is added once for every value of the index. Here that is 5 times, so ∑k=153=5⋅3=15\sum_{k=1}^{5} 3 = 5 \cdot 3 = 15. In general, ∑k=mnc=c(n−m+1)\sum_{k=m}^{n} c = c(n - m + 1).

Related Terms