Question : Find the average of the first 2578 even numbers.
Correct Answer 2579
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 2578 even numbers are
2, 4, 6, 8, . . . . 2578 th terms
Calculation of the sum of the first 2578 even numbers
We can find the sum of the first 2578 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 2578 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 2578 even number,
n = 2578, a = 2, and d = 2
Thus, sum of the first 2578 even numbers
S2578 = 2578/2 [2 × 2 + (2578 – 1) 2]
= 2578/2 [4 + 2577 × 2]
= 2578/2 [4 + 5154]
= 2578/2 × 5158
= 2578/2 × 5158 2579
= 2578 × 2579 = 6648662
⇒ The sum of the first 2578 even numbers (S2578) = 6648662
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 2578 even numbers
= 25782 + 2578
= 6646084 + 2578 = 6648662
⇒ The sum of the first 2578 even numbers = 6648662
Calculation of the Average of the first 2578 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2578 even numbers
= Sum of the first 2578 even numbers/2578
= 6648662/2578 = 2579
Thus, the average of the first 2578 even numbers = 2579 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 2578 even numbers = 2578 + 1 = 2579
Thus, the average of the first 2578 even numbers = 2579 Answer
Similar Questions
(1) What is the average of the first 1364 even numbers?
(2) Find the average of even numbers from 10 to 1194
(3) Find the average of the first 2128 odd numbers.
(4) Find the average of the first 2304 even numbers.
(5) What is the average of the first 943 even numbers?
(6) What is the average of the first 31 even numbers?
(7) Find the average of even numbers from 4 to 486
(8) Find the average of even numbers from 6 to 1294
(9) Find the average of the first 2849 odd numbers.
(10) Find the average of the first 1313 odd numbers.