Average
MCQs Math


Question:     Find the average of the first 3948 even numbers.


Correct Answer  3949

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 3948 even numbers are

2, 4, 6, 8, . . . . 3948 th terms

Calculation of the sum of the first 3948 even numbers

We can find the sum of the first 3948 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 3948 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 3948 even number,

n = 3948, a = 2, and d = 2

Thus, sum of the first 3948 even numbers

S3948 = 3948/2 [2 × 2 + (3948 – 1) 2]

= 3948/2 [4 + 3947 × 2]

= 3948/2 [4 + 7894]

= 3948/2 × 7898

= 3948/2 × 7898 3949

= 3948 × 3949 = 15590652

⇒ The sum of the first 3948 even numbers (S3948) = 15590652

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 3948 even numbers

= 39482 + 3948

= 15586704 + 3948 = 15590652

⇒ The sum of the first 3948 even numbers = 15590652

Calculation of the Average of the first 3948 even numbers

Formula to find the Average

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

Thus, The average of the first 3948 even numbers

= Sum of the first 3948 even numbers/3948

= 15590652/3948 = 3949

Thus, the average of the first 3948 even numbers = 3949 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 3948 even numbers = 3948 + 1 = 3949

Thus, the average of the first 3948 even numbers = 3949 Answer


Similar Questions

(1) Find the average of the first 2727 even numbers.

(2) Find the average of the first 2828 odd numbers.

(3) Find the average of even numbers from 4 to 178

(4) Find the average of even numbers from 10 to 1568

(5) Find the average of odd numbers from 7 to 173

(6) What will be the average of the first 4556 odd numbers?

(7) What is the average of the first 557 even numbers?

(8) Find the average of the first 2195 odd numbers.

(9) What will be the average of the first 4381 odd numbers?

(10) Find the average of the first 3418 even numbers.


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