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Average - Arithmetic


Calculation of average of even or odd numbers


Important rules

Rule: (a) Average of n odd numbers = n

Rule: (b) Average of n even numbers = n + 1

(1) Calculate the average of first 5 odd numbers.

Solution :

We know that, Numbers which are not divisible by 2 are called Odd numbers.

Thus, list of first 5 odd numbers are

1, 3, 5, 7, 9

Now, we know that average of given numbers

Thus, average of first 5 odd numbers

=25/5 =5

Thus, average of first 5 odd numbers = 5 Answer

Shortcut method

Average of first 1 odd numbers = 1

Average of first 2 odd numbers

Average of first 3 odd numbers

Thus, average of first 5 odd numbers = 5 Answer

(2) Find the average of first 11 odd numbers

Solution

Numbers which are not divisible by 2 are called Odd numbers.

Thus, list of first 11 odd numbers is 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21

Sum of first 11 odd numbers

= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

= 121

Thus, Average of first 11 odd numbers =121/11 =11

Alternate Method

We have, List of first 11 odd numbers is 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21

[While finding average the sum of given numbers is required. The sum of given numbers can be obtained by adding them. But it is easy to find the sum of series of numbers using formula. Since the list of 11 odd numbers form an Arithmetic series. Thus using formula to find the sum of n terms of Arithmetic progression should be applied here to find the sum easily. ]

We know that, the sum of n term of AP (Sn)

Where, n = numbers of terms, a = first term and l = last terms

Here we have, n = 11

a = 1 and l = 21

Thus, sum of first 11 odd numbers

= 11 × 11 = 121

Thus, sum of given list of numbers = 121

Now, Average of given numbers

=121/11 = 11

Thus, average of first 11 odd numbers = 11 Answer

Shortcut method/trick to solve the question

Average of first 1 odd numbers = 1

Average of first 2 odd numbers

Average of first 3 odd numbers

Thus, average of first 11 odd numbers = 11 Answer

(3) Find the average of first 50 odd numbers

Solution

Numbers which are not divisible by 2 are called Odd numbers.

Thus, List of continuous odd numbers forms a series which is

1, 3, 5, 7, . . . . . . .

Series which has difference between two continuous terms is equal are called Arithmetic seris.

The sum of n terms of an Arithmetic series (Sn)

Where, n = number of terms, a = first term and d = difference between two consecutive terms, i.e. common difference.

In order to find the Average, it is necessary to calculate the sum of given numbers

According to question, the list of first 50 odd numbers

1, 3, 5, . . . . . ., upto 50thterm

Here first term, a = 1

Common differene, d = 2

Number of terms, n = 50

The sum of 50th term

= 25 × [2 + 49 × 2]

=25 × [2+98]

= 25 × 100

Thus, sum of first 50 odd numbers = 2500

Now, we know that, average of given numbers

Thus, Average of first 50 odd numbers

Thus, Average of first 50 odd numbers = 50 Answer

Shortcut Method/Trick method

Average of first 1 odd numbers = 1

Average of first 2 odd numbers

Average of first 3 odd numbers

Thus, average of first 50 odd numbers = 50 Answer

(4) Find the average of first 100 odd numbers

Solution

Numbers which are not divisible by 2 are called Odd Numbers

Thus, the list of first 100 odd numbers

= 1, 3, 5, 7, . . . . . upto 100th term

Here, this series form an Arithmetic series

In this series , first term (a) = 1

Common difference (d) = 2

And number of term = 100

Thus, sum of n terms (Sn)

= 50 [2 + (99 × 2)]

= 50 (2 + 198)

= 50 × 200

⇒ Sum of 100 terms (S100) = 10000

Now, we know that, Average of given numbers

Thus, Average of first 100 odd numbers =10000/100

= 100

∴ Average of first 100 odd numbers = 100 Answer

Shortcut Method/Trick method

Average of first 1 odd numbers = 1

Average of first 2 odd numbers

Average of first 3 odd numbers

Thus, average of first 100 odd numbers = 100 Answer

(5) Calculate the average of first 1000 odd numbers.

Solution

Numbers which are not divisible by 2 are called Odd Numbers

Thus, the list of first 1000 odd numbers

= 1, 3, 5, 7, . . . . . upto 1000th term

Here, this series form an Arithmetic series

In this series , first term (a) = 1

Common difference (d) = 2

And number of terms (n) = 1000

We know that, sum of n terms (Sn)

Thus, sum of 1000th term

= 500 [2 + (999 × 2)]

= 500 (2 + 1998)

= 500 × 2000

⇒ Sum of 1000 terms (S1000) = 1000000

Now, we know that, Average of given numbers

Thus, Average of first 1000 odd numbers =1000000/1000

= 1000

∴ Average of first 1000 odd numbers = 1000 Answer

Shortcut Method/Trick method

Average of first 1 odd numbers = 1

Average of first 2 odd numbers

Average of first 3 odd numbers

Thus, average of first 1000 odd numbers = 1000 Answer

(6) Find the average of first 5 even numbers

Solution

Numbers divisible by 2 are called even numbers

Thus, list of first 5 even numbers

2, 4, 6, 8, 10

Now, we know that, Average

Thus, Average of given numbers

=30/5 = 6

Thus, Average of first 5 even numbers = 6 Answer

Shortcut Method/Trick to solve the question

Average of first 2 even numbers = (2+4)2= 3

Average of first 3 even numbers = (2+4+6)/3=4

Average of first 4 even numbers

Clearly, Average of first n even numbers = n + 1

Therefore, average of first 5 even numbers = 5+1 = 6 Answer

(7) Find the average of first 10 even numbers.

Solution

Numbers divisible by 2 are called even numbers

Thus, list of first 10 even numbers

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Now, we know that, Average

Sum of first 10 even numbers

= 2+4+6+8+10+12+14+16+18+20

= 110

Thus, Average of first 10 even numbers =110/10=11

Thus, Average of first 10 even numbers = 11 Answer

Shortcut Method/Trick to solve the question

Average of first 2 even numbers = (2+4)2= 3

Average of first 3 even numbers = (2+4+6)/3=4

Average of first 4 even numbers =(2+4+6+8)/4=5

Clearly, Average of first n even numbers = n + 1

Therefore, average of first 10 even numbers = 10+1 = 11 Answer

(8) Calculate the average of first 15 even numbers.

Solution :

Numbers divisible by 2 are called even numbers. For example 2, 4, 6, 8, etc. are even numbers.

Thus, List of first 15 even numbers is

2, 4, 6, 8, . . . . . . ., 30

Since the difference between consecutive numbers of the list is same, thus list of consecutive even numbers forms an Arithmetic Series.

Here, first term (a) = 2

Common difference (d) [difference between consecutive numbers]= 2

Last term (l) = 30

Number of terms (n) 15

We know that, sum of n terms (Sn)

Sum of this series of first 15 even numbers (S15)

= 15 × 16 = 240

Thus, sum of first 15 even numbers = 240

Now, we know that, Average

Thus, Average of first 15 even numbers =240/15 =16

Thus, Average of first 15 even numbers = 16 Answer

Alternate Method

Thus, List of first 15 even numbers is

2, 4, 6, 8, . . . . . . ., 30

Or, 2(1, 2, 3, . . . . . , 15)

Thus, Sum of first 15 even numbers

= 2 ( 1 + 2 + 3 + . . . . . + 15)

Now, We know that, sum of n natural numbers

And Average

Thus, Average of first 15 even numbers

Thus, Average of first 15 even numbers = 16 Answer

Shortcut Method/Trick to solve the question

Average of first 2 even numbers = (2+4)2= 3

Average of first 3 even numbers

Average of first 4 even numbers

Clearly, Average of first n even numbers = n + 1

Therefore, average of first 15 even numbers = 16 + 1 = 16 Answer

 

(9) Find the average of first 50 even numbers.

Solution :

Numbers divisible by 2 are called even numbers. For example 2, 4, 6, 8, etc. are even numbers.

Thus, List of first 15 even numbers is

2, 4, 6, 8, . . . . . . ., 100

Since the difference between consecutive numbers of the list is same, thus list of consecutive even numbers forms an Arithmetic Series.

Here, first term (a) = 2

Common difference (d) [difference between consecutive numbers]= 2

Last term (l) = 100

Number of terms (n) 50

We know that, sum of n terms (Sn)

Sum of this series of first 50 even numbers (S50)

= 25 × 102

= 2550

Thus, sum of first 50 even numbers = 2550

Now, we know that, Average

Thus, Average of first 50 even numbers

Thus, Average of first 50 even numbers = 51 Answer

Alternate Method

Thus, List of first 50 even numbers is

2, 4, 6, 8, . . . . . . ., 100

Or, 2(1, 2, 3, . . . . . , 50)

Thus, Sum of first 50 even numbers

= 2 ( 1 + 2 + 3 + . . . . . + 50)

Now, We know that, sum of n natural numbers

And Average

Thus, Average of first 50 even numbers

Thus, Average of first 50 even numbers = 51 Answer

Shortcut Method/Trick to solve the question

Average of first 2 even numbers = (2+4)2= 3

Average of first 3 even numbers

Average of first 4 even numbers

Clearly, Average of first n even numbers = n + 1

Therefore, average of first 50 even numbers = 50 + 1 = 51 Answer

(10) Find the average of first 150 even numbers.

Solution :

Numbers divisible by 2 are called even numbers. For example 2, 4, 6, 8, etc. are even numbers.

Thus, List of first 150 even numbers is

2, 4, 6, 8, . . . . . . ., 300

Since the difference between consecutive numbers of the list is same, thus list of consecutive even numbers forms an Arithmetic Series.

Here, first term (a) = 2

Common difference (d) [difference between consecutive numbers]= 2

Last term (l) = 300

Number of terms (n) 150

We know that, sum of n terms (Sn)

Sum of this series of first 150 even numbers (S150)

= 75 × 302

= 22650

Thus, sum of first 150 even numbers = 22650

Now, we know that, Average

Thus, Average of first 150 even numbers

Thus, Average of first 150 even numbers = 151 Answer

Alternate Method

Thus, List of first 150 even numbers is

2, 4, 6, 8, . . . . . . ., 300

Or, 2(1, 2, 3, . . . . . , 150)

Thus, Sum of first 150 even numbers

= 2 ( 1 + 2 + 3 + . . . . . + 150)

Now, We know that, sum of n natural numbers

And Average

Thus, Average of first 150 even numbers

Thus, Average of first 150 even numbers = 151 Answer

Shortcut Method/Trick to solve the question

Average of first 2 even numbers = (2+4)2= 3

Average of first 3 even numbers

Average of first 4 even numbers

Clearly, Average of first n even numbers = n + 1

Therefore, average of first 150 even numbers = 150 + 1 = 151 Answer

(11) Find the average of first 5000 even numbers.

Solution :

Numbers divisible by 2 are called even numbers. For example 2, 4, 6, 8, etc. are even numbers.

Thus, List of first 5000 even numbers is

2, 4, 6, 8, . . . . . . ., 10000

Since the difference between consecutive numbers of the list is same, thus list of consecutive even numbers forms an Arithmetic Series.

Here, first term (a) = 2

Common difference (d) [difference between consecutive numbers]= 2

Last term (l) = 10000

Number of terms (n) 5000

We know that, sum of n terms (Sn)

Sum of this series of first 5000 even numbers (S5000)

= 2500 × 10002

= 25005000

Thus, sum of first 5000 even numbers = 25005000

Now, we know that, Average

Thus, Average of first 5000 even numbers

Thus, Average of first 5000 even numbers = 5001 Answer

Alternate Method

Thus, List of first 5000 even numbers is

2, 4, 6, 8, . . . . . . ., 10000

Or, 2(1, 2, 3, . . . . . , 5000)

Thus, Sum of first 5000 even numbers

= 2 ( 1 + 2 + 3 + . . . . . + 5000)

Now, We know that, sum of n natural numbers

And Average

Thus, Average of first 5000 even numbers

Thus, Average of first 5000 even numbers = 5001 Answer

Shortcut Method/Trick to solve the question

Average of first 2 even numbers = (2+4)2= 3

Average of first 3 even numbers

Average of first 4 even numbers

Clearly, Average of first n even numbers = n + 1

Therefore, average of first 5000 even numbers = 5000 + 1 = 5001 Answer




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