Question:
What is the average of the first 951 even numbers?
Correct Answer
952
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 951 even numbers are
2, 4, 6, 8, . . . . 951 th terms
Calculation of the sum of the first 951 even numbers
We can find the sum of the first 951 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 951 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 951 even number,
n = 951, a = 2, and d = 2
Thus, sum of the first 951 even numbers
S951 = 951/2 [2 × 2 + (951 – 1) 2]
= 951/2 [4 + 950 × 2]
= 951/2 [4 + 1900]
= 951/2 × 1904
= 951/2 × 1904 952
= 951 × 952 = 905352
⇒ The sum of the first 951 even numbers (S951) = 905352
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 951 even numbers
= 9512 + 951
= 904401 + 951 = 905352
⇒ The sum of the first 951 even numbers = 905352
Calculation of the Average of the first 951 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 951 even numbers
= Sum of the first 951 even numbers/951
= 905352/951 = 952
Thus, the average of the first 951 even numbers = 952 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 951 even numbers = 951 + 1 = 952
Thus, the average of the first 951 even numbers = 952 Answer
Similar Questions
(1) Find the average of the first 2915 odd numbers.
(2) Find the average of odd numbers from 11 to 1245
(3) Find the average of odd numbers from 13 to 1295
(4) Find the average of odd numbers from 5 to 1211
(5) Find the average of odd numbers from 11 to 1239
(6) Find the average of the first 3990 even numbers.
(7) Find the average of odd numbers from 15 to 1791
(8) Find the average of the first 3709 odd numbers.
(9) Find the average of the first 3238 odd numbers.
(10) Find the average of even numbers from 8 to 1142