Question:
Find the average of the first 2022 even numbers.
Correct Answer
2023
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 the given numbers
The first 2022 even numbers are
2, 4, 6, 8, . . . . 2022 th terms
Calculation of the sum of the first 2022 even numbers
We can find the sum of the first 2022 even 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 2022 even 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 the first 2022 even number,
n = 2022, a = 2, and d = 2
Thus, sum of the first 2022 even numbers
S2022 = 2022/2 [2 × 2 + (2022 – 1) 2]
= 2022/2 [4 + 2021 × 2]
= 2022/2 [4 + 4042]
= 2022/2 × 4046
= 2022/2 × 4046 2023
= 2022 × 2023 = 4090506
⇒ The sum of the first 2022 even numbers (S2022) = 4090506
Shortcut Method to find the sum of the first n even numbers
Thus, the sum of the first n even numbers = n2 + n
Thus, the sum of the first 2022 even numbers
= 20222 + 2022
= 4088484 + 2022 = 4090506
⇒ The sum of the first 2022 even numbers = 4090506
Calculation of the Average of the first 2022 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2022 even numbers
= Sum of the first 2022 even numbers/2022
= 4090506/2022 = 2023
Thus, the average of the first 2022 even numbers = 2023 Answer
Shortcut Trick to find the Average of the first n even numbers
(1) The average of the first 2 even numbers
= 2 + 4/2
= 6/2 = 3
Thus, the average of the first 2 even numbers = 3
(2) The average of the first 3 even numbers
= 2 + 4 + 6/3
= 12/3 = 4
Thus, the average of the first 3 even numbers = 4
(3) The average of the first 4 even numbers
= 2 + 4 + 6 + 8/4
= 20/4 = 5
Thus, the average of the first 4 even numbers = 5
(4) The average of the first 5 even numbers
= 2 + 4 + 6 + 8 + 10/5
= 30/5 = 6
Thus, the average of the first 5 even numbers = 6
Thus, the Average of the First n even numbers = n + 1
Thus, the average of the first 2022 even numbers = 2022 + 1 = 2023
Thus, the average of the first 2022 even numbers = 2023 Answer
Similar Questions
(1) Find the average of the first 3177 even numbers.
(2) Find the average of the first 3394 even numbers.
(3) Find the average of the first 2196 odd numbers.
(4) Find the average of the first 3502 odd numbers.
(5) Find the average of even numbers from 6 to 1708
(6) Find the average of the first 3051 odd numbers.
(7) Find the average of even numbers from 4 to 1704
(8) Find the average of the first 4305 even numbers.
(9) Find the average of even numbers from 4 to 112
(10) Find the average of the first 2933 even numbers.