Question : Find the average of the first 927 odd numbers.
Correct Answer 927
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 927 odd numbers are
1, 3, 5, 7, 9, . . . . 927 th terms
Calculation of the sum of the first 927 odd numbers
We can find the sum of the first 927 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 927 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 927 odd number,
n = 927, a = 1, and d = 2
Thus, sum of the first 927 odd numbers
S927 = 927/2 [2 × 1 + (927 – 1) 2]
= 927/2 [2 + 926 × 2]
= 927/2 [2 + 1852]
= 927/2 × 1854
= 927/2 × 1854 927
= 927 × 927 = 859329
⇒ The sum of first 927 odd numbers (Sn) = 859329
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 927 odd numbers
= 9272 = 859329
⇒ The sum of first 927 odd numbers = 859329
Calculation of the Average of the first 927 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 927 odd numbers
= Sum of first 927 odd numbers/927
= 859329/927 = 927
Thus, the average of the first 927 odd numbers = 927 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 927 odd numbers = 927
Thus, the average of the first 927 odd numbers = 927 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 399
(2) Find the average of odd numbers from 11 to 367
(3) Find the average of even numbers from 8 to 254
(4) Find the average of the first 3788 even numbers.
(5) Find the average of odd numbers from 11 to 289
(6) Find the average of the first 4990 even numbers.
(7) Find the average of even numbers from 8 to 436
(8) Find the average of even numbers from 4 to 216
(9) What is the average of the first 351 even numbers?
(10) What is the average of the first 173 even numbers?