Question : Find the average of the first 542 odd numbers.
Correct Answer 542
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 542 odd numbers are
1, 3, 5, 7, 9, . . . . 542 th terms
Calculation of the sum of the first 542 odd numbers
We can find the sum of the first 542 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 542 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 542 odd number,
n = 542, a = 1, and d = 2
Thus, sum of the first 542 odd numbers
S542 = 542/2 [2 × 1 + (542 – 1) 2]
= 542/2 [2 + 541 × 2]
= 542/2 [2 + 1082]
= 542/2 × 1084
= 542/2 × 1084 542
= 542 × 542 = 293764
⇒ The sum of first 542 odd numbers (Sn) = 293764
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 542 odd numbers
= 5422 = 293764
⇒ The sum of first 542 odd numbers = 293764
Calculation of the Average of the first 542 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 542 odd numbers
= Sum of first 542 odd numbers/542
= 293764/542 = 542
Thus, the average of the first 542 odd numbers = 542 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 542 odd numbers = 542
Thus, the average of the first 542 odd numbers = 542 Answer
Similar Questions
(1) Find the average of the first 2785 even numbers.
(2) What is the average of the first 1936 even numbers?
(3) What is the average of the first 1725 even numbers?
(4) Find the average of even numbers from 10 to 1796
(5) Find the average of even numbers from 4 to 1352
(6) What is the average of the first 1914 even numbers?
(7) Find the average of even numbers from 6 to 58
(8) What is the average of the first 114 odd numbers?
(9) Find the average of the first 2416 even numbers.
(10) What is the average of the first 1205 even numbers?