Question : Find the average of the first 4019 even numbers.
Correct Answer 4020
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 the given numbers
The first 4019 even numbers are
2, 4, 6, 8, . . . . 4019 th terms
Calculation of the sum of the first 4019 even numbers
We can find the sum of the first 4019 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 4019 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 4019 even number,
n = 4019, a = 2, and d = 2
Thus, sum of the first 4019 even numbers
S4019 = 4019/2 [2 × 2 + (4019 – 1) 2]
= 4019/2 [4 + 4018 × 2]
= 4019/2 [4 + 8036]
= 4019/2 × 8040
= 4019/2 × 8040 4020
= 4019 × 4020 = 16156380
⇒ The sum of the first 4019 even numbers (S4019) = 16156380
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 4019 even numbers
= 40192 + 4019
= 16152361 + 4019 = 16156380
⇒ The sum of the first 4019 even numbers = 16156380
Calculation of the Average of the first 4019 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 4019 even numbers
= Sum of the first 4019 even numbers/4019
= 16156380/4019 = 4020
Thus, the average of the first 4019 even numbers = 4020 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 4019 even numbers = 4019 + 1 = 4020
Thus, the average of the first 4019 even numbers = 4020 Answer
Similar Questions
(1) What will be the average of the first 4800 odd numbers?
(2) Find the average of the first 410 odd numbers.
(3) Find the average of the first 2417 even numbers.
(4) Find the average of even numbers from 12 to 804
(5) What is the average of the first 1673 even numbers?
(6) Find the average of the first 858 odd numbers.
(7) Find the average of odd numbers from 7 to 449
(8) Find the average of even numbers from 10 to 566
(9) What is the average of the first 871 even numbers?
(10) Find the average of odd numbers from 13 to 831