Average
MCQs Math


Question:     Find the average of the first 1355 odd numbers.


Correct Answer  1355

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 given numbers

The first 1355 odd numbers are

1, 3, 5, 7, 9, . . . . 1355 th terms

Calculation of the sum of the first 1355 odd numbers

We can find the sum of the first 1355 odd 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 1355 odd 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 first 1355 odd number,

n = 1355, a = 1, and d = 2

Thus, sum of the first 1355 odd numbers

S1355 = 1355/2 [2 × 1 + (1355 – 1) 2]

= 1355/2 [2 + 1354 × 2]

= 1355/2 [2 + 2708]

= 1355/2 × 2710

= 1355/2 × 2710 1355

= 1355 × 1355 = 1836025

⇒ The sum of first 1355 odd numbers (Sn) = 1836025

Shortcut Method to find the sum of first n odd numbers

Thus, the sum of first n odd numbers = n2

Thus, the sum of first 1355 odd numbers

= 13552 = 1836025

⇒ The sum of first 1355 odd numbers = 1836025

Calculation of the Average of the first 1355 odd numbers

Formula to find the Average

Average = Sum of given numbers/Number of numbers

Thus, The average of the first 1355 odd numbers

= Sum of first 1355 odd numbers/1355

= 1836025/1355 = 1355

Thus, the average of the first 1355 odd numbers = 1355 Answer

Shortcut Trick to find the Average of the first n odd numbers

The average of the first 2 odd numbers

= 1 + 3/2

= 4/2 = 2

Thus, the average of the first 2 odd numbers = 2

The average of the first 3 odd numbers

= 1 + 3 + 5/3

= 9/3 = 3

Thus, the average of the first 3 odd numbers = 3

The average of the first 4 odd numbers

= 1 + 3 + 5 + 7/4

= 16/4 = 4

Thus, the average of the first 4 odd numbers = 4

The average of the first 5 odd numbers

= 1 + 3 + 5 + 7 + 9/5

= 25/5 = 5

Thus, the average of the first 5 odd numbers = 5

Thus, the Average of the the First n odd numbers = n

Thus, the average of the first 1355 odd numbers = 1355

Thus, the average of the first 1355 odd numbers = 1355 Answer


Similar Questions

(1) Find the average of the first 3825 even numbers.

(2) Find the average of the first 2287 even numbers.

(3) Find the average of the first 2848 even numbers.

(4) Find the average of odd numbers from 15 to 29

(5) What will be the average of the first 4604 odd numbers?

(6) Find the average of odd numbers from 7 to 563

(7) Find the average of the first 1071 odd numbers.

(8) Find the average of the first 2630 odd numbers.

(9) What will be the average of the first 4452 odd numbers?

(10) What is the average of the first 166 even numbers?


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