Question : What is the average of the first 39 odd numbers?
Correct Answer 39
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 39 odd numbers are
1, 3, 5, 7, 9, . . . . 39 th terms
Caclulation of the sum of the first 39 odd numbers
We can find the sum of first 39 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 given 39 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 39 odd number,
n = 39, a = 1, and d = 2
Thus, sum of the first 39 odd numbers
S39 = 39/2 [2 × 1 + (39 – 1) 2]
= 39/2 [2 + 38 × 2]
= 39/2 [2 + 76]
= 39/2 × 78
= 39/2 × 78 39
= 39 × 39 = 1369
⇒ The sum of first 39 odd numbers (Sn) = 1369
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 39 odd numbers
= 392 = 1521
⇒ The sum of first 39 odd numbers = 1521
Calculation of the Average of the first 39 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 39 odd numbers
= Sum of first 39 odd numbers/39
= 1521/39 = 39
Thus, the average of the first 39 odd numbers = 39 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 39 odd numbers = 39
Thus, the average of the first 39 odd numbers = 39 Answer
Similar Questions
(1) What will be the average of the first 4401 odd numbers?
(2) Find the average of the first 1964 odd numbers.
(3) Find the average of the first 2474 even numbers.
(4) What is the average of the first 531 even numbers?
(5) Find the average of even numbers from 10 to 504
(6) Find the average of the first 3258 odd numbers.
(7) Find the average of the first 823 odd numbers.
(8) Find the average of odd numbers from 15 to 1093
(9) Find the average of odd numbers from 11 to 661
(10) Find the average of the first 2755 even numbers.