Average
MCQs Math


Question:     Find the average of the first 3697 even numbers.


Correct Answer  3698

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 3697 even numbers are

2, 4, 6, 8, . . . . 3697 th terms

Calculation of the sum of the first 3697 even numbers

We can find the sum of the first 3697 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 3697 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 3697 even number,

n = 3697, a = 2, and d = 2

Thus, sum of the first 3697 even numbers

S3697 = 3697/2 [2 × 2 + (3697 – 1) 2]

= 3697/2 [4 + 3696 × 2]

= 3697/2 [4 + 7392]

= 3697/2 × 7396

= 3697/2 × 7396 3698

= 3697 × 3698 = 13671506

⇒ The sum of the first 3697 even numbers (S3697) = 13671506

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 3697 even numbers

= 36972 + 3697

= 13667809 + 3697 = 13671506

⇒ The sum of the first 3697 even numbers = 13671506

Calculation of the Average of the first 3697 even numbers

Formula to find the Average

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

Thus, The average of the first 3697 even numbers

= Sum of the first 3697 even numbers/3697

= 13671506/3697 = 3698

Thus, the average of the first 3697 even numbers = 3698 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 3697 even numbers = 3697 + 1 = 3698

Thus, the average of the first 3697 even numbers = 3698 Answer


Similar Questions

(1) Find the average of odd numbers from 11 to 905

(2) Find the average of the first 3398 odd numbers.

(3) What is the average of the first 1155 even numbers?

(4) Find the average of odd numbers from 13 to 29

(5) Find the average of the first 2362 odd numbers.

(6) Find the average of even numbers from 10 to 306

(7) What is the average of the first 1916 even numbers?

(8) Find the average of the first 1348 odd numbers.

(9) Find the average of the first 2433 odd numbers.

(10) Find the average of even numbers from 10 to 728


NCERT Solution and CBSE Notes for class twelve, eleventh, tenth, ninth, seventh, sixth, fifth, fourth and General Math for competitive Exams. ©