Average
MCQs Math


Question:     Find the average of the first 4856 even numbers.


Correct Answer  4857

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 4856 even numbers are

2, 4, 6, 8, . . . . 4856 th terms

Calculation of the sum of the first 4856 even numbers

We can find the sum of the first 4856 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 4856 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 4856 even number,

n = 4856, a = 2, and d = 2

Thus, sum of the first 4856 even numbers

S4856 = 4856/2 [2 × 2 + (4856 – 1) 2]

= 4856/2 [4 + 4855 × 2]

= 4856/2 [4 + 9710]

= 4856/2 × 9714

= 4856/2 × 9714 4857

= 4856 × 4857 = 23585592

⇒ The sum of the first 4856 even numbers (S4856) = 23585592

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 4856 even numbers

= 48562 + 4856

= 23580736 + 4856 = 23585592

⇒ The sum of the first 4856 even numbers = 23585592

Calculation of the Average of the first 4856 even numbers

Formula to find the Average

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

Thus, The average of the first 4856 even numbers

= Sum of the first 4856 even numbers/4856

= 23585592/4856 = 4857

Thus, the average of the first 4856 even numbers = 4857 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 4856 even numbers = 4856 + 1 = 4857

Thus, the average of the first 4856 even numbers = 4857 Answer


Similar Questions

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(2) Find the average of the first 3174 odd numbers.

(3) What is the average of the first 128 odd numbers?

(4) What is the average of the first 1522 even numbers?

(5) Find the average of even numbers from 10 to 760

(6) What is the average of the first 355 even numbers?

(7) Find the average of the first 878 odd numbers.

(8) Find the average of the first 3217 odd numbers.

(9) Find the average of the first 3748 even numbers.

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