Question : Find the average of the first 2267 odd numbers.
Correct Answer 2267
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 2267 odd numbers are
1, 3, 5, 7, 9, . . . . 2267 th terms
Calculation of the sum of the first 2267 odd numbers
We can find the sum of the first 2267 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 2267 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 2267 odd number,
n = 2267, a = 1, and d = 2
Thus, sum of the first 2267 odd numbers
S2267 = 2267/2 [2 × 1 + (2267 – 1) 2]
= 2267/2 [2 + 2266 × 2]
= 2267/2 [2 + 4532]
= 2267/2 × 4534
= 2267/2 × 4534 2267
= 2267 × 2267 = 5139289
⇒ The sum of first 2267 odd numbers (Sn) = 5139289
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 2267 odd numbers
= 22672 = 5139289
⇒ The sum of first 2267 odd numbers = 5139289
Calculation of the Average of the first 2267 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 2267 odd numbers
= Sum of first 2267 odd numbers/2267
= 5139289/2267 = 2267
Thus, the average of the first 2267 odd numbers = 2267 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 2267 odd numbers = 2267
Thus, the average of the first 2267 odd numbers = 2267 Answer
Similar Questions
(1) Find the average of the first 4743 even numbers.
(2) Find the average of odd numbers from 7 to 911
(3) Find the average of odd numbers from 7 to 1121
(4) Find the average of odd numbers from 7 to 407
(5) Find the average of the first 2406 odd numbers.
(6) Find the average of the first 3167 even numbers.
(7) Find the average of the first 2840 even numbers.
(8) Find the average of the first 799 odd numbers.
(9) Find the average of the first 2021 even numbers.
(10) Find the average of the first 4340 even numbers.