Average of numbers formula | Average of a series of numbers| Exercise – 2

Formulas to find average of numbers | The average of a group of numbers

In this Exercise – 2  given formulas with examples for average of numbers like first “n “natural numbers,  average of even numbers, average of odd numbers, average of constitutive numbers, Average of cubes of first ” n” natural, even, odd numbers … etc

In Exercise – I   page,  given different formulas in “ Averages “ chapter  – Average Concept , Weighted Average, Average age /weight, Average Speed … etc. This formulas helpful to competitive exams

Formulas on Average of numbers:

1. Average of first ” n”  natural numbers =

2. Average of first ” n”  even numbers  = ( n + 1 )

3. Average of first ” n ” odd numbers = n

4. Average of consecutive numbers =

5. Average of 1 to “n” odd numbers =

6. Average of 1 to “n” even numbers =

7. Average of sum of square of first ” n” natural numbers =

8. Average of sum of square of first “n” even numbers =

9.  Average of sum of square of first “n” odd numbers =

10. Average of cubes of first ” n” natural numbers =

11. Average of cubes of first “n” even natural numbers = 2n ( n +1) ²

12. Average of cubes of first “n” odd natural numbers = n (2n² – 1)

13. Average of first “n “multiple of ” m” =

14. If average of “n₁” observations is “A₁“, and average of “n₂” observations is “A₂“, then

Average of (n₁ –  n₂) observations is

Examples on Average of numbers:

Example -1: Find average of first 20 natural numbers

Solution: Here n = 20 then according to above formula (20+1)/2 = 10.5

Example – 2: Find average of  2, 4 , 6 , …… 60  even numbers

Solution: Here using two types of formulas

If we count even numbers in given sum then  n = 30 and its average  (30 +1) = 31

If we take last even numbers i.e  n = 60 then average is (60 +2) / 2  = 31

Example – 3: Find average of the series of  51, 53, 55, ………99

Solution: Here we know it is series of odd numbers from 51 to 99, then

First find Average of 1 to “99” odd numbers = (99 + 1 )/ 2 = 50

First find Average of 1 to “49” odd numbers = (49 + 1)  / 2 = 25

Now average of numbers from 51, 53, 55, ………99  = 50 + 25 = 75.

Example – 4: Find average of the series of  31, 33, 35, ………99

Solution: Here we know it is series of odd numbers from 31 to 99, then

First find Average first 50 odd numbers ( i.e 1 to 99) = 50

First find Average first 15 odd numbers ( i.e 1 to 29 ) = 15

Now average of numbers from 31, 33, 35 ……. 99  = 50 + 15 = 65.

Example – 5: Find the average of series 1² , 2² , 3², …………. 30²

Solution: Here we know it is series first 30 natural number. So according to above formula

= (30+1) (2×30 +1) /6

= 31 x 61 / 6 = 1891 /6 = 315.17

Example – 6: Find the average of series of numbers  1² , 3² , 5², …………. 99²

Solution: Here we find it is series of squares of odd numbers from 1 to 99

Average of sum of square of first “50” odd numbers ( i.e 1² , 3² , 5², …………. 99² )

=( 4 x 50x 50 – 1 ) / 3 = 9999/3 = 3333

Example – 7: Find the average of series of numbers  8, 64, 216, ……… 27000

Solution: Here we find it is series of cubes of even numbers from 2 to 30

Average of sum of cubes of first “15” even numbers ( i.e 2³ , 4³ , 6³, …………. 30³ ) 2n ( n +1) ²

= 2 x 15 x 16 x 16

= 30 x 256 = 7680

Example – 8: Find the average of series of numbers  9, 18, 27, 36,……. 108.

Solution: Here we identifying, it is series of  “12” multiples for  9 ( 108/9 = 12)

According to above formula n = 12 and m = 9

= 9 x (12+1) / 2 = 9 x 13 /2 = 58.5

Example – 9: First 50 natural numbers average

Solution: Here using above formula n = 50

= 50+1 / 2 = 25.5

———————————————-  X ————————————————

Average Problems | Exercise – I

Average Formulas with Examples | Exercise- III

Topics in Quantitative aptitude math for all types of exams

Shortcut Math Tricks for helpful to improve speed in all calculations