Average
MCQs Math


Question:     Find the average of the first 3524 even numbers.


Correct Answer  3525

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 3524 even numbers are

2, 4, 6, 8, . . . . 3524 th terms

Calculation of the sum of the first 3524 even numbers

We can find the sum of the first 3524 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 3524 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 3524 even number,

n = 3524, a = 2, and d = 2

Thus, sum of the first 3524 even numbers

S3524 = 3524/2 [2 × 2 + (3524 – 1) 2]

= 3524/2 [4 + 3523 × 2]

= 3524/2 [4 + 7046]

= 3524/2 × 7050

= 3524/2 × 7050 3525

= 3524 × 3525 = 12422100

⇒ The sum of the first 3524 even numbers (S3524) = 12422100

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 3524 even numbers

= 35242 + 3524

= 12418576 + 3524 = 12422100

⇒ The sum of the first 3524 even numbers = 12422100

Calculation of the Average of the first 3524 even numbers

Formula to find the Average

Average = Sum of the given numbers/Number of the numbers

Thus, The average of the first 3524 even numbers

= Sum of the first 3524 even numbers/3524

= 12422100/3524 = 3525

Thus, the average of the first 3524 even numbers = 3525 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 3524 even numbers = 3524 + 1 = 3525

Thus, the average of the first 3524 even numbers = 3525 Answer


Similar Questions

(1) Find the average of the first 1526 odd numbers.

(2) Find the average of even numbers from 10 to 416

(3) Find the average of even numbers from 4 to 1162

(4) Find the average of even numbers from 12 to 1146

(5) Find the average of even numbers from 10 to 490

(6) Find the average of even numbers from 10 to 1256

(7) What will be the average of the first 4513 odd numbers?

(8) Find the average of odd numbers from 15 to 989

(9) Find the average of odd numbers from 15 to 627

(10) Find the average of odd numbers from 15 to 233


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