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    You are at:Home»Quantitative Aptitude»Average of numbers formula | Average of a series of numbers| Exercise – 2
    Average of numbers formula | Average of first " n" natural,even,odd numbers | Average of 1 to "n" even, odd numbers | Average of sum of square of first "n" natural,even,odd numbers | Average of cubes of first " n" natural,even,odd numbers | Average of first "n "multiple of " m" | The average of a group of numbers.
    Quantitative Aptitude

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

    sivaalluriBy sivaalluriMay 29, 2018Updated:February 24, 2025No Comments5 Mins Read

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    • Formulas to find the average of numbers | The average of a group of numbers
      • Formulas on Average of numbers:
      • Examples of average numbers:

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

    Formulas on Average of numbers:

    1. Average of first ” n”  natural numbers =  \frac{n + 1}{2}

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

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

    4. Average of consecutive numbers =  \frac{\text{First number} + \text{Last number}}{2}

    5. Average of 1 to “n” odd numbers =  \frac{\text{Last odd number} + 1}{2}

    6. Average of 1 to “n” even numbers =  \frac{\text{Last even number} + 2}{2}

    7. Average of the sum of the square of first ” n” natural numbers =  \frac{(n+1) (2n +1)}{6}

    8. Average of a sum of squares of first “n” even numbers =  \frac{2 (n+1)(2n+1)}{3}

    9.  Average of sum of square of first “n” odd numbers =  \frac{4n^2 - 1}{3}

    10. Average of cubes of first ” n” natural numbers =  \frac{n (n+1)^2}{4}

    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” =  \frac{m (n+1)}{2}

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

    Average of (n1 –  n2) observations is  \frac{n_{1}A_{1} - n_{2}A_{2}}{n_{1} - n_{2}}

    Examples of average numbers:

    Example -1: Find the average of the 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 a 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 the average of the series of  31, 33, 35, ………99

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

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

    Find the 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 a series first 30 natural numbers. So according to the above formula

    = (30+1) (2×30 +1) /6

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

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

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

    Average of the sum of squares 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 a series of numbers  8, 64, 216, ……… 27000

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

    Average of the 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 a series of numbers  9, 18, 27, 36,……. 108.

    Solution: Here we identify, it is a 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 the above formula n = 50

    = 50+1 / 2 = 25.5

    Average of numbers formula | Average of first " n" natural,even,odd numbers | Average of 1 to "n" even, odd numbers | Average of sum of square of first "n" natural,even,odd numbers | Average of cubes of first " n" natural,even,odd numbers | Average of first "n "multiple of " m" | The average of a group of numbers.

    ———————————————-  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

    Average age /weight Average Speed averages series of numbers
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