Question : What will be the average of the first 4646 odd numbers?
Correct Answer 4646
Solution & 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 4646 odd numbers are
1, 3, 5, 7, 9, . . . . 4646 th terms
Calculation of the sum of the first 4646 odd numbers
We can find the sum of the first 4646 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 4646 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 4646 odd number,
n = 4646, a = 1, and d = 2
Thus, sum of the first 4646 odd numbers
S4646 = 4646/2 [2 × 1 + (4646 – 1) 2]
= 4646/2 [2 + 4645 × 2]
= 4646/2 [2 + 9290]
= 4646/2 × 9292
= 4646/2 × 9292 4646
= 4646 × 4646 = 21585316
⇒ The sum of first 4646 odd numbers (Sn) = 21585316
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 4646 odd numbers
= 46462 = 21585316
⇒ The sum of first 4646 odd numbers = 21585316
Calculation of the Average of the first 4646 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 4646 odd numbers
= Sum of first 4646 odd numbers/4646
= 21585316/4646 = 4646
Thus, the average of the first 4646 odd numbers = 4646 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 4646 odd numbers = 4646
Thus, the average of the first 4646 odd numbers = 4646 Answer
Similar Questions
(1) What is the average of the first 1778 even numbers?
(2) What will be the average of the first 4957 odd numbers?
(3) Find the average of the first 3272 even numbers.
(4) Find the average of the first 1225 odd numbers.
(5) Find the average of the first 3142 even numbers.
(6) What is the average of the first 141 even numbers?
(7) Find the average of the first 2558 odd numbers.
(8) Find the average of odd numbers from 15 to 1197
(9) Find the average of the first 2990 odd numbers.
(10) Find the average of odd numbers from 11 to 885