Question:
What will be the average of the first 4262 odd numbers?
Correct Answer
4262
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 4262 odd numbers are
1, 3, 5, 7, 9, . . . . 4262 th terms
Calculation of the sum of the first 4262 odd numbers
We can find the sum of the first 4262 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 4262 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 4262 odd number,
n = 4262, a = 1, and d = 2
Thus, sum of the first 4262 odd numbers
S4262 = 4262/2 [2 × 1 + (4262 – 1) 2]
= 4262/2 [2 + 4261 × 2]
= 4262/2 [2 + 8522]
= 4262/2 × 8524
= 4262/2 × 8524 4262
= 4262 × 4262 = 18164644
⇒ The sum of first 4262 odd numbers (Sn) = 18164644
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 4262 odd numbers
= 42622 = 18164644
⇒ The sum of first 4262 odd numbers = 18164644
Calculation of the Average of the first 4262 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 4262 odd numbers
= Sum of first 4262 odd numbers/4262
= 18164644/4262 = 4262
Thus, the average of the first 4262 odd numbers = 4262 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 4262 odd numbers = 4262
Thus, the average of the first 4262 odd numbers = 4262 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1542
(2) Find the average of even numbers from 12 to 130
(3) Find the average of odd numbers from 15 to 1611
(4) Find the average of the first 2047 odd numbers.
(5) What will be the average of the first 4752 odd numbers?
(6) Find the average of the first 3173 odd numbers.
(7) Find the average of the first 4166 even numbers.
(8) Find the average of even numbers from 12 to 794
(9) Find the average of odd numbers from 13 to 509
(10) What will be the average of the first 4506 odd numbers?