Difference between revisions of "Summation"
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*<math>\sum_{i=0}^{n} {n\choose i} = 2^n</math> | *<math>\sum_{i=0}^{n} {n\choose i} = 2^n</math> | ||
*<math>\sum_{i,j}^{n} = \sum_i^n \sum_j^n</math> | *<math>\sum_{i,j}^{n} = \sum_i^n \sum_j^n</math> | ||
+ | |||
+ | *<math>\sum_{i=0}^{2n} {(x^2 \times 10^i)}=(\sum_{j=0}^n {(3x \times 10^j)})^2 + \sum_{k=0}^n {(2x^2 \times 10^k)}</math> | ||
+ | Or | ||
+ | *<math>x^2\sum_{i=0}^{2n} {10^i}=(x \sum_{j=0}^n {(3 \times 10^j)})^2 + x^2\sum_{k=0}^n {(2 \times 10^k)}</math> | ||
+ | Look for PaperMath’s sums if you want to look deeper into these identities | ||
== Problems == | == Problems == |
Latest revision as of 15:39, 8 October 2023
A summation is the sum of a number of terms (addends). Summations are often written using sigma notation .
Contents
Definition
For , and a set (or any other algebraic structure), . Here refers to the index of summation, is the lower bound, and is the upper bound.
As an example, . Note that if , then the sum is .
Quite often, sigma notation is used in a slightly different format to denote certain sums. For example, refers to a cyclic sum, and refers to all subsets which are in . Usually, the bottom of the sigma contains a logical condition, as in , where the sum only includes the terms which divide into .
Identities
- , and in general
- , and in general
Or
Look for PaperMath’s sums if you want to look deeper into these identities
Problems
Introductory
- Evaluate the following sums:
Intermediate
- The nine horizontal and nine vertical lines on an checkerboard form rectangles, of which are squares. The number can be written in the form where and are relatively prime positive integers. Find (1997 AIME, #2)