Average
MCQs Math


Question:     What is the average of the first 26 even numbers?


Correct Answer  27

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 26 even numbers are

2, 4, 6, 8, . . . . 26 th terms

Caclulation of the sum of the first 26 even numbers

We can find the sum of first 26 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 given 26 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 26 even number,

n = 26, a = 2, and d = 2

Thus, sum of the first 26 even numbers

S26 = 26/2 [2 × 2 + (26 – 1) 2]

= 13 [4 + 25 × 2]

= 13 [4 + 50]

= 13 × 54 = 702

⇒ The sum of the first 26 even numbers (Sn) = 702

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 26 even numbers

= 262 + 26

= 676 + 26 = 702

⇒ The sum of the first 26 even numbers (Sn) = 600

Calculation of the Average of the first 26 even numbers

Formula to find the Average

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

Thus, The average of the first 26 even numbers

= Sum of the first 26 even numbers/26

= 702/26 = 27

Thus, the average of the first 26 even numbers = 27 Answer

Shortcut Trick to find the Average of the first n even numbers

The average of the first 2 even numbers

= 2 + 4/2

= 6/2 = 3

Thus, the average of the first 2 even numbers = 3

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

The average of the first 4 even numbers

= 2 + 4 + 6 + 8/4

= 16/4 = 5

Thus, the average of the first 4 even numbers = 5

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 26 even numbers = 26 + 1 = 27

Thus, the average of the first 26 even numbers = 27 Answer


Similar Questions

(1) Find the average of the first 3641 odd numbers.

(2) Find the average of the first 3924 odd numbers.

(3) Find the average of the first 3091 even numbers.

(4) What is the average of the first 892 even numbers?

(5) Find the average of odd numbers from 15 to 763

(6) Find the average of the first 3608 even numbers.

(7) What is the average of the first 1034 even numbers?

(8) Find the average of odd numbers from 11 to 1349

(9) Find the average of odd numbers from 11 to 651

(10) Find the average of even numbers from 10 to 1314


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