Question:
What is the average of the first 275 even numbers?
Correct Answer
276
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 275 even numbers are
2, 4, 6, 8, . . . . 275 th terms
Calculation of the sum of the first 275 even numbers
We can find the sum of the first 275 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 275 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 275 even number,
n = 275, a = 2, and d = 2
Thus, sum of the first 275 even numbers
S275 = 275/2 [2 × 2 + (275 – 1) 2]
= 275/2 [4 + 274 × 2]
= 275/2 [4 + 548]
= 275/2 × 552
= 275/2 × 552 276
= 275 × 276 = 75900
⇒ The sum of the first 275 even numbers (S275) = 75900
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 275 even numbers
= 2752 + 275
= 75625 + 275 = 75900
⇒ The sum of the first 275 even numbers = 75900
Calculation of the Average of the first 275 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 275 even numbers
= Sum of the first 275 even numbers/275
= 75900/275 = 276
Thus, the average of the first 275 even numbers = 276 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 275 even numbers = 275 + 1 = 276
Thus, the average of the first 275 even numbers = 276 Answer
Similar Questions
(1) Find the average of the first 2002 odd numbers.
(2) Find the average of odd numbers from 13 to 1043
(3) Find the average of the first 4817 even numbers.
(4) Find the average of the first 4727 even numbers.
(5) Find the average of the first 3423 even numbers.
(6) Find the average of odd numbers from 3 to 665
(7) What is the average of the first 1581 even numbers?
(8) Find the average of the first 4691 even numbers.
(9) Find the average of odd numbers from 11 to 987
(10) Find the average of the first 4139 even numbers.