Question:
Find the average of the first 2114 odd numbers.
Correct Answer
2114
Solution And 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 2114 odd numbers are
1, 3, 5, 7, 9, . . . . 2114 th terms
Calculation of the sum of the first 2114 odd numbers
We can find the sum of the first 2114 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 2114 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 2114 odd number,
n = 2114, a = 1, and d = 2
Thus, sum of the first 2114 odd numbers
S2114 = 2114/2 [2 × 1 + (2114 – 1) 2]
= 2114/2 [2 + 2113 × 2]
= 2114/2 [2 + 4226]
= 2114/2 × 4228
= 2114/2 × 4228 2114
= 2114 × 2114 = 4468996
⇒ The sum of first 2114 odd numbers (Sn) = 4468996
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 2114 odd numbers
= 21142 = 4468996
⇒ The sum of first 2114 odd numbers = 4468996
Calculation of the Average of the first 2114 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 2114 odd numbers
= Sum of first 2114 odd numbers/2114
= 4468996/2114 = 2114
Thus, the average of the first 2114 odd numbers = 2114 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 2114 odd numbers = 2114
Thus, the average of the first 2114 odd numbers = 2114 Answer
Similar Questions
(1) Find the average of odd numbers from 3 to 739
(2) What is the average of the first 1337 even numbers?
(3) Find the average of the first 1879 odd numbers.
(4) Find the average of even numbers from 6 to 476
(5) What is the average of the first 1261 even numbers?
(6) Find the average of the first 4849 even numbers.
(7) Find the average of even numbers from 12 to 508
(8) Find the average of odd numbers from 11 to 691
(9) What is the average of the first 508 even numbers?
(10) Find the average of even numbers from 6 to 374