Average
MCQs Math


Question:     Find the average of the first 3030 even numbers.


Correct Answer  3031

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 3030 even numbers are

2, 4, 6, 8, . . . . 3030 th terms

Calculation of the sum of the first 3030 even numbers

We can find the sum of the first 3030 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 3030 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 3030 even number,

n = 3030, a = 2, and d = 2

Thus, sum of the first 3030 even numbers

S3030 = 3030/2 [2 × 2 + (3030 – 1) 2]

= 3030/2 [4 + 3029 × 2]

= 3030/2 [4 + 6058]

= 3030/2 × 6062

= 3030/2 × 6062 3031

= 3030 × 3031 = 9183930

⇒ The sum of the first 3030 even numbers (S3030) = 9183930

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 3030 even numbers

= 30302 + 3030

= 9180900 + 3030 = 9183930

⇒ The sum of the first 3030 even numbers = 9183930

Calculation of the Average of the first 3030 even numbers

Formula to find the Average

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

Thus, The average of the first 3030 even numbers

= Sum of the first 3030 even numbers/3030

= 9183930/3030 = 3031

Thus, the average of the first 3030 even numbers = 3031 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 3030 even numbers = 3030 + 1 = 3031

Thus, the average of the first 3030 even numbers = 3031 Answer


Similar Questions

(1) Find the average of the first 1220 odd numbers.

(2) Find the average of the first 4505 even numbers.

(3) Find the average of the first 1916 odd numbers.

(4) Find the average of the first 4570 even numbers.

(5) Find the average of the first 1661 odd numbers.

(6) Find the average of even numbers from 10 to 758

(7) Find the average of even numbers from 10 to 1316

(8) Find the average of even numbers from 4 to 1960

(9) Find the average of the first 4338 even numbers.

(10) Find the average of odd numbers from 15 to 1631


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