Sigma Notation

Sigma notation offers a concise method for representing the sum of a sequence of numbers. The uppercase Greek letter sigma acts as a shorthand for "the sum of."

The basic structure of sigma is:

i=abf(i)

This means: “Sum the values of f(i) as it ranges from a to b.”

Components of the basic structure and what they mean:

 = summation symbol
i = index of the summation, and it is usually a variable
a = lower limit, which is the start value
b = the upper limit, which is the end value
f(i) = expression or function to sum
For example: i=25i = 2 + 3 + 4 + 5 = 14
 

Sigma notation helps us work with sums easily and in a compact form. Understanding its properties helps simplify expressions and evaluate sums effectively.

1. When you have a constant multiplied by a function (or a term) within a summation, you can take that constant outside the sigma:
i=mnc. f(i) = c .i=mn f(i)
This means that when every term in the sum shares the same factor, you can factor it out and multiply it by the total sum later.

2. If you're summing a series of terms where each term is a sum (or difference) of two or more functions, you can split that into separate summations:
i=mn(f(i)g(i)) = i=mnf(i)  i=mng(i)
So, instead of adding the functions first and then summing, you can sum each function separately and then add (or subtract) the results.

3. To sum the constant: i=abc = c(b - a + 1)

4. The choice of the index variable is arbitrary. Whether you call the index 'i' or 'j' (or any other letter), the value of the summation won't change.
i=1nf(i) = j=1nf(j)

5. When working with sums or sequences indexed by a variable (like 'n' in your example), you can change the starting or ending point of the index by substituting a new variable. This substitution affects both the index variable within the term and the limits of the summation (if applicable).
i=0nai = k=1n+1ak-1
 

As discussed earlier, the sigma symbol (), is a Greek alphabet. The summation notation that utilizes this symbol is commonly known as a 'series', as it represents a sum.

How to Write Sigma Notation?

So far, we’ve learnt about the sigma notation and what it represents. Now, let’s learn how to write the notation properly with the help of the steps given below: 

1. First, identify the pattern and the general term. If the sequence is arithmetic or geometric, use its general term formula. Otherwise, we have to find the general formula through observation.

2. Choose a lowercase letter for the index, like 'i' or 'k', common for summation.

3. Examine the sequence to determine the first and last values of the index

4. Then, finally, express the sequence using sigma notation.

For example: Write the sigma notation for 3, 6, 9, 12, …, 21
Solution: The given sequence is an arithmetic sequence
First term, a = 3
The common difference, d = 3
Then, 
         an = a + (n - 1)d
         an = 3 + (n - 1)3 = 3n
         n = 21 - 33 + 1 = 7
Finally, i = 173i = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6) + 3(7) = 3 + 6 + 9 + 12 + 15 + 18 + 21 = 84
 

Expanding summation notation helps us see the summation of each individual term and better understand the structure. Let’s now see how to expand summation notation by following these steps:
Replace the index in the general term with each value from first to last, incrementing by 1. Begin by determining the index variable and the range it covers. After knowing its range, substitute each of these values in the equation, one at a time. 
Then place a plus symbol between the resulting terms.
For example: Expand i=36i3
Solution: Given, 
                    i = 3, 4, 5, 6 and in the general term i3
We get, 
          i=36i3 = 33  + 43  + 53  + 63  = 27 + 64 + 125 + 216 = 432
  

What are the Formulas for Sigma Notation?

Not all summations can be simplified using formulas, certain formulas have been derived to simplify specific summation notations. These are useful for finding the sum of natural numbers, squares of natural numbers, cubes of natural numbers, even numbers, odd numbers, and more.

Sigma notation has numerous practical applications in real-world scenarios that involve repeated addition or accumulation. Let’s take a look at some of those applications.

1. In finance and economics, investment returns over time are often modeled using summations. For example, budgeting requires the calculation of total expenses.

2. In engineering, signal processing involves summing signal values over time, while stress and load calculations require summing distributed loads or forces.

3. When analyzing demographics and population, census data is aggregated by summing up the number of people in each region to calculate the total population.

4. In environmental science, to find the air quality or rainfall data over time.

5. In architecture & construction, summation is used to calculate material costing and structural analysis.