Average can be defined as the mean of a certain set of numbers. Questions based on average are seen in various competitive exams especially government job examinations. Candidates must be aware of the method required to solve the questions on Average. We are providing you with the study notes on Average along with examples.

 

 

• The average of first n natural numbers is                            :         (n + 1)/2.
• The average of square of natural numbers till n is              :        ((n + 1)  (2n + 1))/6
• The average of cubes of natural numbers till n is                :         (n (n + 1)²)/4.
• The average of odd numbers from 1 to n is                         :         (last odd number+1)/2.
• The average of even numbers from 1 to n is                        :         (last even number + 2)/2.

Question. What is the average of odd numbers from 1 to 40?

Solution. The required average
=(last odd number+1)/2
=(39+1)/2
= 20

Question . What is the average of even numbers from 1 to 81?

Solution. The required average
=(last even number+2)/2
=(80+2)/2
= 41

• If n is odd: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the middle number.
• If n is even: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the average of the middle two numbers.

• The average of first n consecutive even numbers is                       :     (n + 1)

• The average of first n consecutive odd numbers is                         :       n.

•  The average of squares of first n consecutive even numbers is     :      (2 (n + 1)  (2n + 1))/3.
• The average of squares of consecutive even numbers till n is        :      ((n + 1)  (n + 2))/3.
• The average of squares of consecutive odd numbers till n is         :       (n (n + 2))/3.

Question. Find the average of squares of first 19 consecutive even numbers.

Solution. The required average
=(2 (n+1)(2n+1))/3=(2(19+1)(2×19+1))/3
=(2×20×39)/3=1560/3=520