Question:
Find the average of the first 4512 even numbers.
Correct Answer
4513
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 4512 even numbers are
2, 4, 6, 8, . . . . 4512 th terms
Calculation of the sum of the first 4512 even numbers
We can find the sum of the first 4512 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 4512 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 4512 even number,
n = 4512, a = 2, and d = 2
Thus, sum of the first 4512 even numbers
S4512 = 4512/2 [2 × 2 + (4512 – 1) 2]
= 4512/2 [4 + 4511 × 2]
= 4512/2 [4 + 9022]
= 4512/2 × 9026
= 4512/2 × 9026 4513
= 4512 × 4513 = 20362656
⇒ The sum of the first 4512 even numbers (S4512) = 20362656
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 4512 even numbers
= 45122 + 4512
= 20358144 + 4512 = 20362656
⇒ The sum of the first 4512 even numbers = 20362656
Calculation of the Average of the first 4512 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 4512 even numbers
= Sum of the first 4512 even numbers/4512
= 20362656/4512 = 4513
Thus, the average of the first 4512 even numbers = 4513 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 4512 even numbers = 4512 + 1 = 4513
Thus, the average of the first 4512 even numbers = 4513 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 535
(2) Find the average of even numbers from 6 to 1850
(3) Find the average of even numbers from 10 to 338
(4) Find the average of odd numbers from 13 to 1345
(5) Find the average of even numbers from 10 to 1138
(6) Find the average of even numbers from 6 to 962
(7) Find the average of odd numbers from 9 to 975
(8) Find the average of the first 2779 even numbers.
(9) Find the average of the first 2314 odd numbers.
(10) Find the average of the first 2772 even numbers.