Average
MCQs Math


Question:     Find the average of the first 3068 even numbers.


Correct Answer  3069

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 3068 even numbers are

2, 4, 6, 8, . . . . 3068 th terms

Calculation of the sum of the first 3068 even numbers

We can find the sum of the first 3068 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 3068 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 3068 even number,

n = 3068, a = 2, and d = 2

Thus, sum of the first 3068 even numbers

S3068 = 3068/2 [2 × 2 + (3068 – 1) 2]

= 3068/2 [4 + 3067 × 2]

= 3068/2 [4 + 6134]

= 3068/2 × 6138

= 3068/2 × 6138 3069

= 3068 × 3069 = 9415692

⇒ The sum of the first 3068 even numbers (S3068) = 9415692

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 3068 even numbers

= 30682 + 3068

= 9412624 + 3068 = 9415692

⇒ The sum of the first 3068 even numbers = 9415692

Calculation of the Average of the first 3068 even numbers

Formula to find the Average

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

Thus, The average of the first 3068 even numbers

= Sum of the first 3068 even numbers/3068

= 9415692/3068 = 3069

Thus, the average of the first 3068 even numbers = 3069 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 3068 even numbers = 3068 + 1 = 3069

Thus, the average of the first 3068 even numbers = 3069 Answer


Similar Questions

(1) Find the average of odd numbers from 13 to 471

(2) What will be the average of the first 4309 odd numbers?

(3) Find the average of odd numbers from 9 to 169

(4) What is the average of the first 901 even numbers?

(5) Find the average of odd numbers from 15 to 107

(6) Find the average of odd numbers from 3 to 641

(7) Find the average of even numbers from 4 to 1872

(8) If the average of four consecutive even numbers is 31, then find the smallest and the greatest numbers among the given even numbers.

(9) Find the average of odd numbers from 5 to 407

(10) What is the average of the first 560 even numbers?


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