Average
MCQs Math


Question:     Find the average of the first 4095 even numbers.


Correct Answer  4096

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 4095 even numbers are

2, 4, 6, 8, . . . . 4095 th terms

Calculation of the sum of the first 4095 even numbers

We can find the sum of the first 4095 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 4095 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 4095 even number,

n = 4095, a = 2, and d = 2

Thus, sum of the first 4095 even numbers

S4095 = 4095/2 [2 × 2 + (4095 – 1) 2]

= 4095/2 [4 + 4094 × 2]

= 4095/2 [4 + 8188]

= 4095/2 × 8192

= 4095/2 × 8192 4096

= 4095 × 4096 = 16773120

⇒ The sum of the first 4095 even numbers (S4095) = 16773120

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 4095 even numbers

= 40952 + 4095

= 16769025 + 4095 = 16773120

⇒ The sum of the first 4095 even numbers = 16773120

Calculation of the Average of the first 4095 even numbers

Formula to find the Average

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

Thus, The average of the first 4095 even numbers

= Sum of the first 4095 even numbers/4095

= 16773120/4095 = 4096

Thus, the average of the first 4095 even numbers = 4096 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 4095 even numbers = 4095 + 1 = 4096

Thus, the average of the first 4095 even numbers = 4096 Answer


Similar Questions

(1) Find the average of even numbers from 10 to 768

(2) Find the average of even numbers from 10 to 634

(3) Find the average of the first 1851 odd numbers.

(4) Find the average of even numbers from 4 to 150

(5) Find the average of the first 2337 even numbers.

(6) What is the average of the first 1557 even numbers?

(7) What will be the average of the first 4605 odd numbers?

(8) Find the average of odd numbers from 13 to 377

(9) Find the average of odd numbers from 7 to 1399

(10) Find the average of the first 2045 even numbers.


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