Average
MCQs Math


Question:     Find the average of the first 4491 even numbers.


Correct Answer  4492

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 4491 even numbers are

2, 4, 6, 8, . . . . 4491 th terms

Calculation of the sum of the first 4491 even numbers

We can find the sum of the first 4491 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 4491 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 4491 even number,

n = 4491, a = 2, and d = 2

Thus, sum of the first 4491 even numbers

S4491 = 4491/2 [2 × 2 + (4491 – 1) 2]

= 4491/2 [4 + 4490 × 2]

= 4491/2 [4 + 8980]

= 4491/2 × 8984

= 4491/2 × 8984 4492

= 4491 × 4492 = 20173572

⇒ The sum of the first 4491 even numbers (S4491) = 20173572

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 4491 even numbers

= 44912 + 4491

= 20169081 + 4491 = 20173572

⇒ The sum of the first 4491 even numbers = 20173572

Calculation of the Average of the first 4491 even numbers

Formula to find the Average

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

Thus, The average of the first 4491 even numbers

= Sum of the first 4491 even numbers/4491

= 20173572/4491 = 4492

Thus, the average of the first 4491 even numbers = 4492 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 4491 even numbers = 4491 + 1 = 4492

Thus, the average of the first 4491 even numbers = 4492 Answer


Similar Questions

(1) Find the average of the first 3902 odd numbers.

(2) Find the average of the first 3728 even numbers.

(3) Find the average of odd numbers from 11 to 257

(4) Find the average of the first 2363 even numbers.

(5) Find the average of even numbers from 10 to 1374

(6) Find the average of even numbers from 4 to 1066

(7) Find the average of even numbers from 8 to 1142

(8) What is the average of the first 894 even numbers?

(9) Find the average of the first 1137 odd numbers.

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