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

Contents

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 (2n^{2} – 1)

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

**14.** If average of “n_{1}” observations is “A_{1}“, and average of “n_{2}” observations is “A_{2}“, then

Average of (n_{1} – n_{2}) 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 1^{2} , 2^{2} , 3^{2}, …………. 30^{2}

**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 1^{2} , 3^{2} , 5^{2}, …………. 99^{2}

**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 1^{2} , 3^{2} , 5^{2}, …………. 99^{2 })

=( 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 2^{3} , 4^{3} , 6^{3}, …………. 30^{3 }) 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:** F**irst 50 natural numbers average**

Solution: Here using above formula n = 50

= 50+1 / 2 = 25.5

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*Average Problems* | Exercise – I

__Average Formulas with Examples | Exercise- III
__

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