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Summation

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A summation is the sum of a number of terms (addends). Summations are often written using sigma notation \left(\sum \right).

Contents

Definition

For b\ge a, and a set c (or any other algebraic structure), \sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}. Here i refers to the index of summation, a is the lower bound, and b is the upper bound.

As an example, \sum_{i=3}^6 i^3 = 3^3 + 4^3 + 5^3 + 6^3. Note that if a>b, then the sum is 0.

Quite often, sigma notation is used slightly different format to denote certain sums. For example, \sum_{cyc} refers to a cyclic sum, and \sum_{a,b \in S} refers to all subsets a, b which are in S. Usually, the bottom of the sigma contains a logical condition, as in \sum_{i|n}^{n} i, where the sum only includes the terms i which divide into n.

Identities

  • \sum_{i=a}^{b}(f(i)+g(i))=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}g(i)
  • \sum_{i=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)
  • \sum_{i=1}^{n} i= \frac{n(n+1)}{2}, and in general \sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}
  • \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
  • \sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2
  • \sum_{i=0}^{n} x^n = \frac{x^{n+1}-1}{x-1}, and in general \sum_{i=a}^{b} c^i = \frac{c^{b+1}-c^a}{c-1}
  • \sum_{i=0}^{n} {n\choose i} = 2^n
  • \sum_{i,j}^{n} = \sum_i^n \sum_j^n

Problems

Introductory

  • Evaluate the following sums:
    • \sum_{i=1}^{20} i
    • \sum_{i=5}^{15} i + 1
    • \sum_{i=1}^{9} {10\choose i}

Intermediate

  • The nine horizontal and nine vertical lines on an 8\times8 checkerboard form r rectangles, of which s are squares. The number s/r can be written in the form m/n, where m and n are relatively prime positive integers. Find m + n. (1997 AIME, #2)

Olympiad

See Also

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