Summation Notation
Summation Notation
Simple sums are usually written with plus signs, though as more terms get added, the expression becomes longer. The summation notation is used to condense sums, making them quicker to write.
Isaiah Lopez
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Introduction
Introduction
Summations are used to express the addition of larger quantities. For example, If you wanted to say 1 + 2 + 3 + 4 and so on, all the way up to ten, you could use summation notation. The symbol used to represent summations is the Greek letter sigma. The expression below is equal to the sum of all the integers numbers from 1 to 10.
The n in this expression is called the the index. The number at the bottom of the sigma is the starting value of the index, and the number at the top is the ending value. Since n = 1 is at the bottom and 10 is at the top, the expression represents the sum of the integers from 1 to 10. The third component of a summation is called the summand, which is placed to the right of the sigma. It shows the formula being applied to each step of the index. In the example above, it is simply n, meaning that no changes are made to each step and the numbers are added plainly.
Changing the summand
Changing the summand
If the summand in the equation above were 2n, you would multiply each step of the summation by 2. This summation would then be 2(1) + 2(2) + 2(3) + 2(4), and so on until the value of the index is reaches 10. The summand can be simple or complex. Below is an example where more operations are applied to the index.
In this example, the index begins at 1 and ends at 4, and the summand is (2n)² + 1. This means that the expression represents the sum ((2(1))² + 1) + ((2(2))² + 1) + ((2(3))² + 1) + ((2(4))² + 1), which simplifies to 5 + 17 + 37 + 65, adding up to a final answer of 124.
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