# How to solve Summation problems explained with examples?

Summation is used to jot down a big series or number of phrases in a quick method. It is extensively utilized in mathematics, as in calculus the usage of summation is very important. In this article, we can find out about the definition, primary idea, and the way to calculate summation through a lot of examples.

## What is Summation?

Summation is a procedure of the addition of a chain of any form of numbers. It is including up a sequence of items. A summation is likewise known as sum, is the result of arithmetically including numbers or quantities. A summation continually carries a whole number of phrases.

It is the sequential addition of a set of numbers. It is easy to do for some numbers, however can get greater complicated with fractions, actual numbers, or algebra. It also can sum function, matrices, and vectors.

Summation notation is permitting expression that includes a sum to be expressed in an easy compact manner. It instructs us to sum factors of a chain.

The upper-case Greek letter **sigma** which is written mathematically as **Σ **is used for summation. In simple words, summation notation is **sigma** (**Σ**).

## Rules of Summation

Name | Rules |

Constant Rule | = a |

Sum Rule | + b = + |

Difference Rule | – b = – |

## How to calculate Summation?

The summation can be calculated in two ways, one by a simple method and the other is by using formulas.

Some basic formulas of summation are given below.

- = 1 + 2 + 3 + … + n = n (n + 1)/2
- = 1
^{2}+ 2^{2}+ 3^{2}+ … + n^{2}= n (n + 1) (2n + 1)/6 - = 1
^{3}+ 2^{3}+ 3^{3}+ … + n^{3}= (n (n + 1)/2)^{3}

Let us take some examples to understand how to calculate summation problems. You can also use an online summation Calculator for accurate results.

**Example 1**

Solve – 9

**Solution **

**Step 1:** understand operations.

Values 1 to 7 = 1, 2, 3, 4, 5, 6, 7

Equation = 4x – 9

**Step 2:** put the values in equation one by one.

When **x = 1**

4x – 9 = 4(1) – 9= 4 – 9 = **-5**

When **x = 2**

4x – 9 = 4(2) – 9 = 8 – 9 = **-1**

When **x = 3**

4x – 9 = 4(3) – 9 = 12 – 9 = **3**

When **x = 4**

4x – 9 = 4(4) – 9 = 16 – 9 = **7**

When **x = 5**

4x – 9 = 4(5) – 9 = 20 – 9 = **11**

When **x = 6**

4x – 9 = 4(6) – 9 = 24 – 9 = **15**

When **x = 7**

4x – 9 = 4(7) – 9 = 28 – 9 = **19**

**Step 3:** add all the results.

– 5 – 1 + 3 + 7 + 11 + 15 + 19

– 9 = 49

**Alternate method**

**Step 1: **apply difference law

– 9 = –

**Step 2:** take constant outside the summation notation.

– 9 = 4 – 9

**Step 3:** Take general formula

= 1 + 2 + 3 + … + n = n (n + 1)/2

**Step 4:** Apply formula.

– 9 = 4(7(7 + 1)/2) – 9 (7)

= 14 (8) – 9 (7)

= 112 – 63 = 49

**Example 2**

Solve – 20

**Solution **

**Step 1:** understand operations.

Values 1 to 9 = 1, 2, 3, 4, 5, 6, 7, 8, 9

Equation = 5x^{2} – 20

**Step 2:** put the values in equation one by one.

When **x = 1**

5x^{2} – 20 = 5(1)^{2} – 20 = 5(1) – 20 = 5 – 20 = **-15**

When **x = 2**

5x^{2} – 20 = 5(2)^{2} – 20 = 5(4) – 20 = 20 – 20 = **0**

When **x = 3**

5x^{2} – 20 = 5(3)^{2} – 20 = 5(9) – 20 = 45 – 20 = **25**

When **x = 4**

5x^{2} – 20 = 5(4)^{2} – 20 = 5(16) – 20 = 80 – 20 = **60**

When **x = 5**

5x^{2} – 20 = 5(5)^{2} – 20 = 5(25) – 20 = 125 – 20 = **105**

When **x = 6**

5x^{2} – 20 = 5(6)^{2} – 20 = 5(36) – 20 = 180 – 20 = **160**

When **x = 7**

5x^{2} – 20 = 5(7)^{2} – 20 = 5(49) – 20 = 245 – 20 = **225**

When **x = 8**

5x^{2} – 20 = 5(8)^{2} – 20 = 5(64) – 20 = 320 – 20 = **300**

When **x = 9**

5x^{2} – 20 = 5(9)^{2} – 20 = 5(81) – 20 = 405 – 20 = **385**

**Step 3:** add all the results.

-15 + 0 + 25 + 60 + 105 + 160 + 225 + 300 + 385 = **1245**

– 20 = 1245

**Alternate method**

**Step 1: **apply difference law

– 20 = –

**Step 2:** take constant outside the summation notation.

– 20 = 5 – 20

**Step 3:** Take general formula

= 1^{2} + 2^{2} + 3^{2} + … + n^{2} = n (n + 1) (2n + 1)/6

**Step 4:** apply formula.

– 20 = 5(9(9 +1) (2(9) + 1)/6) – 20 (9)

= 5(9(10) (18 + 1)/6) – 20 (9)

= 5(90 (19)/6) – 180

= 5(15 (19)) – 180

= 5(285) – 180

= 1425 – 180

= 1245

## Applications of Summation

Mathematics isn’t always actual life; however, it could be implemented to actual life, normally through simplified approximations. Summation notation saves one having to write out a listing of phrases of a sequence and could be very convenient.

However, it handiest applies while the phrases of the collection have an easy formula. In theory, it’s far essential. If I write 2 + 3 + 5 + …, you would possibly suppose I am writing a sequence that continues 2 + 3 + 5 + 8 + 12 + 17 + … (wherein the distinction among successive phrases will increase by 1 every time), once I become surely questioning of 1 + 3 + 5 + 7 + 11 + 13 + …, the collection of top numbers.

There isn’t any manner to inform from a given set of phrases what the overall rule is. Summation notation makes this clear. As this series is simply written as .

## Summary

Summation is used to reduce the difficulty of writing a large number of sequences most simply and easily. Summation is denoted by sigma written as **Σ. **

Now you are witnessed that this topic is not difficult. Just a little effort is required. Once you practice it with your own hands you can easily grab the whole knowledge about summation.