Question : Find the average of the first 2812 even numbers.
Correct Answer 2813
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 2812 even numbers are
2, 4, 6, 8, . . . . 2812 th terms
Calculation of the sum of the first 2812 even numbers
We can find the sum of the first 2812 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 2812 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 2812 even number,
n = 2812, a = 2, and d = 2
Thus, sum of the first 2812 even numbers
S2812 = 2812/2 [2 × 2 + (2812 – 1) 2]
= 2812/2 [4 + 2811 × 2]
= 2812/2 [4 + 5622]
= 2812/2 × 5626
= 2812/2 × 5626 2813
= 2812 × 2813 = 7910156
⇒ The sum of the first 2812 even numbers (S2812) = 7910156
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 2812 even numbers
= 28122 + 2812
= 7907344 + 2812 = 7910156
⇒ The sum of the first 2812 even numbers = 7910156
Calculation of the Average of the first 2812 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2812 even numbers
= Sum of the first 2812 even numbers/2812
= 7910156/2812 = 2813
Thus, the average of the first 2812 even numbers = 2813 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 2812 even numbers = 2812 + 1 = 2813
Thus, the average of the first 2812 even numbers = 2813 Answer
Similar Questions
(1) What is the average of the first 706 even numbers?
(2) Find the average of the first 2169 even numbers.
(3) Find the average of odd numbers from 13 to 601
(4) Find the average of the first 3474 even numbers.
(5) Find the average of even numbers from 6 to 1540
(6) What is the average of the first 403 even numbers?
(7) Find the average of the first 2997 even numbers.
(8) Find the average of the first 4982 even numbers.
(9) Find the average of odd numbers from 9 to 449
(10) What is the average of the first 1363 even numbers?