AVERAGE OF FIRST N NATURAL NUMBERS

Formula to find the sum of first n natural numbers :

Example 1 :

Find the average of first 100 natural numbers.

Solution :

= ⁽¹⁰⁰ ⁺ ¹⁾⁄₂

= ⁵¹⁄₂

= 25.5

Example 2 :

Find the average of first 50 positive integers which are multiples of 7.

Solution :

Average of first 50 positive integers which are multiples of 7 :

= 7 ⋅ 25.5

= 178.5

Example 3 :

Find the average of first 75 positive integers which are divsible by 3.

Solution :

Average of first 75 positive integers which are divsible by 3 :

= 3 ⋅ 38

= 114

Example 4 :

Find the average of all natural numbers from 51 to 112.

Solution :

The natural numbers from 51 to 112 are

51, 52, 53, ......., 112

Number of natural numbers from 51 to 112 :

= 112 - 50

= 62

Average of all natural numbers from 51 to 112 :

Example 5 :

Find the average of all natural numbers less than or equal to 1000, which are evenly divsible by 11.

Solution :

First let us find the numer of natural numbers less than or equal to 1000, which are divisible by 11.

= ¹⁰⁰⁰⁄₁₁

= 90.9090.......

Numer of natural numbers less than or equal to 1000, which are divisible by 11 :

= largest integer contained in 90.9090.......

= 90

Average of all natural numbers less than or equal to 1000, which are evenly divsible by 11 :

= 11 ⋅ 45.5

= 500.5

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