In daily life most of the time we need some average related methods, but the methods we used all are different. Our parents and grandparents do the numeric calculations orally and fast than us.

When, we buy something we there need average calculating methods. Not only on paper but also in real life we require average finding methods. Then what is mean by average?

**Average:**

When we have taken some group of numbers, they may be correlated or not but we have to find their average. Then we have to take sum of all numbers and have to divide it by the quantity of numbers.

Hence, mathematically average can be defined as,

**Average = sum of all quantities / number of quantities**

For example:

We have taken numbers like 2, 3, 5, 7, 8 and 10. These numbers are not consecutive numbers and they are total six numbers.

Hence we can find the average of these numbers as follows:

Average = sum of all numbers / quantity of number

Average = (2 + 3 + 5 + 7 + 8 + 10) / 6 = 35 / 6

Average = 5.83

In this way, we can find the average of numbers which are not consecutive.

__When numbers are consecutive:__

1.) If we have taken the first 9 natural numbers as 1, 2, 3, 4, 5, 6, 7, 8, 9. These are 9 in number.

Hence, average can be calculated as

Average = (1 + 2 + 3+ 4 + 5 + 6 + 7+ 8+ 9) / 9

Average = 45 / 9 = 5

Thus, we got the average as 9.

This is the general method of finding average of any group of numbers.

Alternative method for consecutive numbers:

1) Here, the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are the consecutive numbers, then their average will be the middle number.

Here, total 9 numbers and 5^{th} number is the exactly middle number and it is 5.

Thus by seeing only we can find the average of consecutive numbers.

2) If we have taken the consecutive numbers as 13, 14, 15, 16, 17, 18. All these numbers are total six.

Generally, average = (13 + 14 + 15 + 16 + 17 + 18) / 6

Average = 93/ 5

Average = 15 .5

But, by alternative method, these are total six number then middle number is between 3^{rd} and 4^{th} number. Third number is 15 and fourth number is 16 hence the middle of them is 15.5. Hence, here the average will be 15.5.

Average calculating method for consecutive odd numbers:

1) If we have to find the average of first six consecutive odd numbers which are 1, 3, 5, 7, 9, and 11.

Then by general method,

Average = (1 + 3 + 5 + 7 + 9 + 11) / 6

Average = 36 / 6

Average = 6

__Shortcut method:__

- The average of first one odd number, average = 1
- The average offirst two odd numbers 1 and 3 is,

Average = (1 + 3)/ 2 = 4 /2

Thus, average of first two consecutive odd numbers is 2.

- The average of first three consecutive odd numbers 1, 3, 5, is

Average = (1 + 3 + 5) / 3 = 9/ 3= 3

Thus, average of first three odd consecutive numbers is 3.

Similarly,

2) The average of first four odd consecutive numbers is 4.

3) The average of first five odd consecutive numbers is 5.

4) And hence, the average of first six odd consecutive numbers is 6.

5) If we have to find the average of first 10 odd consecutive numbers then directly we say that the average will be 10.

6) If we have to find the average of first 50 odd consecutive numbers then the average will be 50.

7) Average of first 100 odd consecutive numbers = 100

8) Average of first 1000 odd consecutive numbers = 1000

Hence,

__In general, the average of first n odd consecutive numbers = n__

Average calculating method for consecutive even numbers:

1) The average of first six even consecutive numbers 2, 4, 6, 8, 10 and 12 is given by,

Average = (2 + 4 + 6+ 8 + 10 + 12) / 6

Average = 42/ 6 = 7

- The average of first one even number 2 is,

Average = 2

- The average of first two even numbers 2 and 4 is,

Average = (2 +4) / 2 = 6/ 2 = 3

- The average of first three even numbers 2, 4, 6 is,

Average = (2 + 4 + 6) / 3 = 4

Similarly,

- The average of first four even numbers 2, 4, 6, 8 is,

Average = 5

- The average of first five even numbers 2, 4, 6, 8, 10 is,

Average = 6

Hence,

**In general, the average of first n consecutive even numbers is**

** (n + 1)**

Thus,

- The average of first 50 even numbers is = 50 + 1 = 51
- The average of first 100 even numbers is = 100 + 1 = 101
- The average of first 1000 even numbers is = 1000 + 1 = 1001