Summation Formulas

When a large number of data are given, and sometimes sum total of the values is required. Then summation is needed here. Many times we need to calculate many terms of a sequence. Therefore methods for summation of a series are very important in mathematics. To write a very large number, summation notation is very useful.  In simple words, summation notation helps to write a short form for the addition of a very large number of data sequences. In this topic, we will discuss the summation formulas with examples. Let us learn it!

What is Summation?

A summation i.e. a sum is the result of arithmetically adding all numbers or quantities given in the form of sequence. A summation always contains an integral number of terms. There can be as few as two terms, or as many as a thousand or even more. Some summations contain infinitely many terms.

For this reason, the summation symbol was devised i.e. \(\Sigma\)

\(\large x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+……..x_{n}=\sum_{i-n}^{n}x_{i}\)

In this section we will need to do a brief review of summation notation or sigma notation. We will start out with two integers, n and m, with n < m and a list of numbers denoted as follows,

\({a_n},\,\,{a_{n + 1}},\,\,{a_{n + 2}},\,\, \ldots ,\,\,{a_{m – 2}},\,\,{a_{m – 1}},\,\,{a_m}\)

\({a_n} + \,{a_{n + 1}} + \,\,{a_{n + 2}} + \,\, \ldots  + \,\,{a_{m – 2}} + \,\,{a_{m – 1}} + \,\,{a_m}\)

For large lists this can be a complex notation. The case above is represented as follows.

\(\sum \limits_{i = n} ^m {{a_i}}  = {a_n} + \,{a_{n + 1}} + \,\, {a_{n + 2}} + \,\, \ldots  + \,\,{a_{m – 2}} + \,\,{a_{m – 1}} + \,\,{a_m}\)

The i is called the index of summation. This notation tells us to add all the a_i’s given.

Source:en.wikipedia.org

Important Summation Formulas

1. \(\sum\limits_{i\, = \,{i_{\,0}}}^n {c{a_i}} = c\sum\limits_{i\, = \,{i_{\,0}}}^n {{a_i}}\) where c is any number.
2. \(\sum\limits_{i\, = \,{i_{\,0}}}^n {\left( {{a_i} \pm {b_i}} \right)} = \sum\limits_{i\, = \,{i_{\,0}}}^n {{a_i}}  \pm \sum \limits_{i\, = \,{i_{\,0}}}^n {{b_i}}\)
3. \(\displaystyle \sum \limits_{i = 1}^n c = cn\\\)
4. \(\displaystyle \sum \limits_{i = 1}^n i = \frac{{n\left( {n + 1} \right)}}{2}\\\)
5. \(\displaystyle \sum \limits_{i = 1}^n {{i^2}} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\\\)
6. \(\displaystyle \sum \limits_{i = 1}^n {{i^3}} = { \left[ {\frac{{n \left( {n + 1} \right)}}{2}} \right]^2}\\\)

Solved Examples for Summation Formulas

Q.1: Using the formulas find the summation value of: \(sum\limits_{i = 1}^{100} {{{\left( {3 – 2i} \right)}^2}}\)

Solution: The first thing that we will do square out the stuff being summed. Then we may break up the summation using the properties. Thus we have,

\(\begin {align*}\sum \limits_{i = 1}^{100} {{{( {3 – 2i} )}^2}} & = \sum\limits_{i = 1}^{100} { 9 – 12i + 4{i^2}} \\  &  = \sum\limits_{i = 1}^{100} 9  – \sum\limits_{i = 1}^{100} {12i}  + \sum\limits_{i =  1}^{100} {4{i^2}} \\ &  = \sum\limits_{i = 1}^{100} 9  – 12 \sum \limits_{i = 1}^{100} i  +  4\sum\limits_{i = 1} ^{100} {{i^2}} \end{align*}\)

Now, using the formulas, sum will be,

\(\begin {align*} \sum\limits_{i = 1}^{100}  {{{ \left( {3 – 2i} \right)}^2}} & = 9\left( {100} \right) – 12\left( {\frac{{100\left( {101} \right)}}{2}} \right) + 4\left( {\frac{{100 \left( { 101 } \right)\left( {201} \right)}}{6}} \right)\\ &  = 1293700\end{align*}\)

Q.2: Find the sum of first 10 odd natural numbers:

Solution: Sequence will be 1,3,5,7,9,11….

It is an A P series. Thus a = 1

d = 2

and n =10

Formula is,

(S = \frac{n}{ 2 }[ 2a+ (n-1) d]\)

Sum of 10 terms will be,

\(S = \frac{10}{2} [2 \times 1 +(10-1) \times 2]\)

S =100

Thus, sum of 10 odd numbers is = 100