Average
MCQs Math


Question:     What is the average of the first 777 even numbers?


Correct Answer  778

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 777 even numbers are

2, 4, 6, 8, . . . . 777 th terms

Calculation of the sum of the first 777 even numbers

We can find the sum of the first 777 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 777 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 777 even number,

n = 777, a = 2, and d = 2

Thus, sum of the first 777 even numbers

S777 = 777/2 [2 × 2 + (777 – 1) 2]

= 777/2 [4 + 776 × 2]

= 777/2 [4 + 1552]

= 777/2 × 1556

= 777/2 × 1556 778

= 777 × 778 = 604506

⇒ The sum of the first 777 even numbers (S777) = 604506

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 777 even numbers

= 7772 + 777

= 603729 + 777 = 604506

⇒ The sum of the first 777 even numbers = 604506

Calculation of the Average of the first 777 even numbers

Formula to find the Average

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

Thus, The average of the first 777 even numbers

= Sum of the first 777 even numbers/777

= 604506/777 = 778

Thus, the average of the first 777 even numbers = 778 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 777 even numbers = 777 + 1 = 778

Thus, the average of the first 777 even numbers = 778 Answer


Similar Questions

(1) Find the average of the first 2944 even numbers.

(2) Find the average of even numbers from 4 to 1092

(3) If the average of three consecutive odd numbers is 23, then which is the greatest among these odd numbers?

(4) What is the average of the first 500 even numbers?

(5) What is the average of the first 1289 even numbers?

(6) Find the average of odd numbers from 13 to 677

(7) If the average of 50 consecutive even numbers is 55, then find the smallest number.

(8) Find the average of the first 3296 odd numbers.

(9) What is the average of the first 1302 even numbers?

(10) Find the average of odd numbers from 5 to 261


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