Question:
Find the average of the first 2028 even numbers.
Correct Answer
2029
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 2028 even numbers are
2, 4, 6, 8, . . . . 2028 th terms
Calculation of the sum of the first 2028 even numbers
We can find the sum of the first 2028 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 2028 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 2028 even number,
n = 2028, a = 2, and d = 2
Thus, sum of the first 2028 even numbers
S2028 = 2028/2 [2 × 2 + (2028 – 1) 2]
= 2028/2 [4 + 2027 × 2]
= 2028/2 [4 + 4054]
= 2028/2 × 4058
= 2028/2 × 4058 2029
= 2028 × 2029 = 4114812
⇒ The sum of the first 2028 even numbers (S2028) = 4114812
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 2028 even numbers
= 20282 + 2028
= 4112784 + 2028 = 4114812
⇒ The sum of the first 2028 even numbers = 4114812
Calculation of the Average of the first 2028 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2028 even numbers
= Sum of the first 2028 even numbers/2028
= 4114812/2028 = 2029
Thus, the average of the first 2028 even numbers = 2029 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 2028 even numbers = 2028 + 1 = 2029
Thus, the average of the first 2028 even numbers = 2029 Answer
Similar Questions
(1) Find the average of the first 1899 odd numbers.
(2) Find the average of even numbers from 12 to 360
(3) Find the average of even numbers from 4 to 1544
(4) Find the average of the first 3249 odd numbers.
(5) Find the average of odd numbers from 7 to 187
(6) Find the average of the first 4522 even numbers.
(7) What is the average of the first 573 even numbers?
(8) Find the average of the first 3916 odd numbers.
(9) Find the average of the first 2121 even numbers.
(10) Find the average of even numbers from 10 to 96