Average
MCQs Math


Question:     Find the average of the first 2513 even numbers.


Correct Answer  2514

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 2513 even numbers are

2, 4, 6, 8, . . . . 2513 th terms

Calculation of the sum of the first 2513 even numbers

We can find the sum of the first 2513 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 2513 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 2513 even number,

n = 2513, a = 2, and d = 2

Thus, sum of the first 2513 even numbers

S2513 = 2513/2 [2 × 2 + (2513 – 1) 2]

= 2513/2 [4 + 2512 × 2]

= 2513/2 [4 + 5024]

= 2513/2 × 5028

= 2513/2 × 5028 2514

= 2513 × 2514 = 6317682

⇒ The sum of the first 2513 even numbers (S2513) = 6317682

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 2513 even numbers

= 25132 + 2513

= 6315169 + 2513 = 6317682

⇒ The sum of the first 2513 even numbers = 6317682

Calculation of the Average of the first 2513 even numbers

Formula to find the Average

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

Thus, The average of the first 2513 even numbers

= Sum of the first 2513 even numbers/2513

= 6317682/2513 = 2514

Thus, the average of the first 2513 even numbers = 2514 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 2513 even numbers = 2513 + 1 = 2514

Thus, the average of the first 2513 even numbers = 2514 Answer


Similar Questions

(1) Find the average of even numbers from 12 to 1514

(2) Find the average of the first 3542 odd numbers.

(3) Find the average of the first 3868 even numbers.

(4) Find the average of odd numbers from 7 to 317

(5) Find the average of the first 1344 odd numbers.

(6) Find the average of odd numbers from 13 to 221

(7) Find the average of odd numbers from 13 to 225

(8) Find the average of odd numbers from 13 to 487

(9) Find the average of odd numbers from 5 to 1193

(10) Find the average of odd numbers from 15 to 1287


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