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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