Average
MCQs Math


Question:     Find the average of the first 4644 even numbers.


Correct Answer  4645

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 4644 even numbers are

2, 4, 6, 8, . . . . 4644 th terms

Calculation of the sum of the first 4644 even numbers

We can find the sum of the first 4644 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 4644 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 4644 even number,

n = 4644, a = 2, and d = 2

Thus, sum of the first 4644 even numbers

S4644 = 4644/2 [2 × 2 + (4644 – 1) 2]

= 4644/2 [4 + 4643 × 2]

= 4644/2 [4 + 9286]

= 4644/2 × 9290

= 4644/2 × 9290 4645

= 4644 × 4645 = 21571380

⇒ The sum of the first 4644 even numbers (S4644) = 21571380

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 4644 even numbers

= 46442 + 4644

= 21566736 + 4644 = 21571380

⇒ The sum of the first 4644 even numbers = 21571380

Calculation of the Average of the first 4644 even numbers

Formula to find the Average

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

Thus, The average of the first 4644 even numbers

= Sum of the first 4644 even numbers/4644

= 21571380/4644 = 4645

Thus, the average of the first 4644 even numbers = 4645 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 4644 even numbers = 4644 + 1 = 4645

Thus, the average of the first 4644 even numbers = 4645 Answer


Similar Questions

(1) What will be the average of the first 4435 odd numbers?

(2) Find the average of even numbers from 12 to 1876

(3) Find the average of odd numbers from 3 to 595

(4) What will be the average of the first 4525 odd numbers?

(5) Find the average of the first 2211 odd numbers.

(6) Find the average of the first 1389 odd numbers.

(7) Find the average of even numbers from 10 to 80

(8) What will be the average of the first 4279 odd numbers?

(9) Find the average of the first 1553 odd numbers.

(10) What is the average of the first 728 even numbers?


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