Question : What is the average of the first 279 even numbers?
Correct Answer 280
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 279 even numbers are
2, 4, 6, 8, . . . . 279 th terms
Calculation of the sum of the first 279 even numbers
We can find the sum of the first 279 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 279 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 279 even number,
n = 279, a = 2, and d = 2
Thus, sum of the first 279 even numbers
S279 = 279/2 [2 × 2 + (279 – 1) 2]
= 279/2 [4 + 278 × 2]
= 279/2 [4 + 556]
= 279/2 × 560
= 279/2 × 560 280
= 279 × 280 = 78120
⇒ The sum of the first 279 even numbers (S279) = 78120
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 279 even numbers
= 2792 + 279
= 77841 + 279 = 78120
⇒ The sum of the first 279 even numbers = 78120
Calculation of the Average of the first 279 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 279 even numbers
= Sum of the first 279 even numbers/279
= 78120/279 = 280
Thus, the average of the first 279 even numbers = 280 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 279 even numbers = 279 + 1 = 280
Thus, the average of the first 279 even numbers = 280 Answer
Similar Questions
(1) What is the average of the first 27 even numbers?
(2) Find the average of even numbers from 12 to 1194
(3) Find the average of odd numbers from 3 to 245
(4) Find the average of odd numbers from 11 to 487
(5) Find the average of even numbers from 10 to 334
(6) What is the average of the first 1490 even numbers?
(7) Find the average of the first 900 odd numbers.
(8) Find the average of odd numbers from 5 to 1179
(9) Find the average of odd numbers from 13 to 1099
(10) Find the average of the first 3914 odd numbers.