Question:
Find the average of the first 4647 even numbers.
Correct Answer
4648
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 4647 even numbers are
2, 4, 6, 8, . . . . 4647 th terms
Calculation of the sum of the first 4647 even numbers
We can find the sum of the first 4647 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 4647 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 4647 even number,
n = 4647, a = 2, and d = 2
Thus, sum of the first 4647 even numbers
S4647 = 4647/2 [2 × 2 + (4647 – 1) 2]
= 4647/2 [4 + 4646 × 2]
= 4647/2 [4 + 9292]
= 4647/2 × 9296
= 4647/2 × 9296 4648
= 4647 × 4648 = 21599256
⇒ The sum of the first 4647 even numbers (S4647) = 21599256
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 4647 even numbers
= 46472 + 4647
= 21594609 + 4647 = 21599256
⇒ The sum of the first 4647 even numbers = 21599256
Calculation of the Average of the first 4647 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 4647 even numbers
= Sum of the first 4647 even numbers/4647
= 21599256/4647 = 4648
Thus, the average of the first 4647 even numbers = 4648 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 4647 even numbers = 4647 + 1 = 4648
Thus, the average of the first 4647 even numbers = 4648 Answer
Similar Questions
(1) Find the average of the first 1319 odd numbers.
(2) Find the average of the first 4611 even numbers.
(3) Find the average of odd numbers from 15 to 1695
(4) Find the average of odd numbers from 15 to 1001
(5) Find the average of the first 1383 odd numbers.
(6) Find the average of the first 2465 even numbers.
(7) Find the average of the first 760 odd numbers.
(8) Find the average of even numbers from 12 to 284
(9) What is the average of the first 1496 even numbers?
(10) What is the average of the first 389 even numbers?