Average
MCQs Math


Question:     Find the average of the first 3571 even numbers.


Correct Answer  3572

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 3571 even numbers are

2, 4, 6, 8, . . . . 3571 th terms

Calculation of the sum of the first 3571 even numbers

We can find the sum of the first 3571 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 3571 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 3571 even number,

n = 3571, a = 2, and d = 2

Thus, sum of the first 3571 even numbers

S3571 = 3571/2 [2 × 2 + (3571 – 1) 2]

= 3571/2 [4 + 3570 × 2]

= 3571/2 [4 + 7140]

= 3571/2 × 7144

= 3571/2 × 7144 3572

= 3571 × 3572 = 12755612

⇒ The sum of the first 3571 even numbers (S3571) = 12755612

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 3571 even numbers

= 35712 + 3571

= 12752041 + 3571 = 12755612

⇒ The sum of the first 3571 even numbers = 12755612

Calculation of the Average of the first 3571 even numbers

Formula to find the Average

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

Thus, The average of the first 3571 even numbers

= Sum of the first 3571 even numbers/3571

= 12755612/3571 = 3572

Thus, the average of the first 3571 even numbers = 3572 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 3571 even numbers = 3571 + 1 = 3572

Thus, the average of the first 3571 even numbers = 3572 Answer


Similar Questions

(1) Find the average of even numbers from 12 to 308

(2) Find the average of the first 1662 odd numbers.

(3) What is the average of the first 443 even numbers?

(4) Find the average of odd numbers from 15 to 1253

(5) Find the average of even numbers from 6 to 264

(6) Find the average of odd numbers from 7 to 849

(7) Find the average of odd numbers from 11 to 791

(8) What will be the average of the first 4353 odd numbers?

(9) Find the average of the first 2322 even numbers.

(10) Find the average of even numbers from 4 to 206


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