Question:
What will be the average of the first 4762 odd numbers?
Correct Answer
4762
Solution And Explanation
Explanation
Method to find the average
Step : (1) Find the sum of given numbers
Step: (2) Divide the sum of given number by the number of numbers. This will give the average of given numbers
The first 4762 odd numbers are
1, 3, 5, 7, 9, . . . . 4762 th terms
Calculation of the sum of the first 4762 odd numbers
We can find the sum of the first 4762 odd numbers by simply adding them, but this is a bit difficult. And if the list is long, it is very difficult to find their sum. So, in such a situation, we will use a formula to find the sum of given numbers that form a particular pattern.
Here, the list of the first 4762 odd numbers forms an Arithmetic series
In an Arithmetic Series, the common difference is the same. This means the difference between two consecutive terms are same in an Arithmetic Series.
The sum of n terms of an Arithmetic Series
Sn = n/2 [2a + (n – 1) d]
Where, n = number of terms, a = first term, and d = common difference
In the series of first 4762 odd number,
n = 4762, a = 1, and d = 2
Thus, sum of the first 4762 odd numbers
S4762 = 4762/2 [2 × 1 + (4762 – 1) 2]
= 4762/2 [2 + 4761 × 2]
= 4762/2 [2 + 9522]
= 4762/2 × 9524
= 4762/2 × 9524 4762
= 4762 × 4762 = 22676644
⇒ The sum of first 4762 odd numbers (Sn) = 22676644
Shortcut Method to find the sum of first n odd numbers
Thus, the sum of first n odd numbers = n2
Thus, the sum of first 4762 odd numbers
= 47622 = 22676644
⇒ The sum of first 4762 odd numbers = 22676644
Calculation of the Average of the first 4762 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 4762 odd numbers
= Sum of first 4762 odd numbers/4762
= 22676644/4762 = 4762
Thus, the average of the first 4762 odd numbers = 4762 Answer
Shortcut Trick to find the Average of the first n odd numbers
The average of the first 2 odd numbers
= 1 + 3/2
= 4/2 = 2
Thus, the average of the first 2 odd numbers = 2
The average of the first 3 odd numbers
= 1 + 3 + 5/3
= 9/3 = 3
Thus, the average of the first 3 odd numbers = 3
The average of the first 4 odd numbers
= 1 + 3 + 5 + 7/4
= 16/4 = 4
Thus, the average of the first 4 odd numbers = 4
The average of the first 5 odd numbers
= 1 + 3 + 5 + 7 + 9/5
= 25/5 = 5
Thus, the average of the first 5 odd numbers = 5
Thus, the Average of the the First n odd numbers = n
Thus, the average of the first 4762 odd numbers = 4762
Thus, the average of the first 4762 odd numbers = 4762 Answer
Similar Questions
(1) Find the average of the first 3517 odd numbers.
(2) Find the average of even numbers from 8 to 1062
(3) What is the average of the first 49 odd numbers?
(4) Find the average of odd numbers from 9 to 931
(5) Find the average of even numbers from 12 to 1626
(6) What is the average of the first 1321 even numbers?
(7) What is the average of the first 25 odd numbers?
(8) What will be the average of the first 4948 odd numbers?
(9) Find the average of even numbers from 10 to 1100
(10) Find the average of the first 3119 even numbers.