Average
MCQs Math


Question:     What is the average of the first 422 even numbers?


Correct Answer  423

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 422 even numbers are

2, 4, 6, 8, . . . . 422 th terms

Calculation of the sum of the first 422 even numbers

We can find the sum of the first 422 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 422 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 422 even number,

n = 422, a = 2, and d = 2

Thus, sum of the first 422 even numbers

S422 = 422/2 [2 × 2 + (422 – 1) 2]

= 422/2 [4 + 421 × 2]

= 422/2 [4 + 842]

= 422/2 × 846

= 422/2 × 846 423

= 422 × 423 = 178506

⇒ The sum of the first 422 even numbers (S422) = 178506

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 422 even numbers

= 4222 + 422

= 178084 + 422 = 178506

⇒ The sum of the first 422 even numbers = 178506

Calculation of the Average of the first 422 even numbers

Formula to find the Average

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

Thus, The average of the first 422 even numbers

= Sum of the first 422 even numbers/422

= 178506/422 = 423

Thus, the average of the first 422 even numbers = 423 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 422 even numbers = 422 + 1 = 423

Thus, the average of the first 422 even numbers = 423 Answer


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