Average
MCQs Math


Question:     What is the average of the first 1132 even numbers?


Correct Answer  1133

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 1132 even numbers are

2, 4, 6, 8, . . . . 1132 th terms

Calculation of the sum of the first 1132 even numbers

We can find the sum of the first 1132 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 1132 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 1132 even number,

n = 1132, a = 2, and d = 2

Thus, sum of the first 1132 even numbers

S1132 = 1132/2 [2 × 2 + (1132 – 1) 2]

= 1132/2 [4 + 1131 × 2]

= 1132/2 [4 + 2262]

= 1132/2 × 2266

= 1132/2 × 2266 1133

= 1132 × 1133 = 1282556

⇒ The sum of the first 1132 even numbers (S1132) = 1282556

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 1132 even numbers

= 11322 + 1132

= 1281424 + 1132 = 1282556

⇒ The sum of the first 1132 even numbers = 1282556

Calculation of the Average of the first 1132 even numbers

Formula to find the Average

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

Thus, The average of the first 1132 even numbers

= Sum of the first 1132 even numbers/1132

= 1282556/1132 = 1133

Thus, the average of the first 1132 even numbers = 1133 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 1132 even numbers = 1132 + 1 = 1133

Thus, the average of the first 1132 even numbers = 1133 Answer


Similar Questions

(1) Find the average of even numbers from 10 to 258

(2) Find the average of the first 1834 odd numbers.

(3) What is the average of the first 111 odd numbers?

(4) Find the average of odd numbers from 9 to 307

(5) Find the average of even numbers from 10 to 76

(6) Find the average of even numbers from 6 to 1818

(7) Find the average of the first 2326 even numbers.

(8) Find the average of odd numbers from 5 to 465

(9) Find the average of the first 4969 even numbers.

(10) Find the average of even numbers from 6 to 742


NCERT Solution and CBSE Notes for class twelve, eleventh, tenth, ninth, seventh, sixth, fifth, fourth and General Math for competitive Exams. ©