Average
MCQs Math


Question:     Find the average of the first 3515 odd numbers.


Correct Answer  3515

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 3515 odd numbers are

1, 3, 5, 7, 9, . . . . 3515 th terms

Calculation of the sum of the first 3515 odd numbers

We can find the sum of the first 3515 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 3515 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 3515 odd number,

n = 3515, a = 1, and d = 2

Thus, sum of the first 3515 odd numbers

S3515 = 3515/2 [2 × 1 + (3515 – 1) 2]

= 3515/2 [2 + 3514 × 2]

= 3515/2 [2 + 7028]

= 3515/2 × 7030

= 3515/2 × 7030 3515

= 3515 × 3515 = 12355225

⇒ The sum of first 3515 odd numbers (Sn) = 12355225

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 3515 odd numbers

= 35152 = 12355225

⇒ The sum of first 3515 odd numbers = 12355225

Calculation of the Average of the first 3515 odd numbers

Formula to find the Average

Average = Sum of given numbers/Number of numbers

Thus, The average of the first 3515 odd numbers

= Sum of first 3515 odd numbers/3515

= 12355225/3515 = 3515

Thus, the average of the first 3515 odd numbers = 3515 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 3515 odd numbers = 3515

Thus, the average of the first 3515 odd numbers = 3515 Answer


Similar Questions

(1) Find the average of the first 362 odd numbers.

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

(3) What will be the average of the first 4248 odd numbers?

(4) Find the average of odd numbers from 5 to 659

(5) Find the average of the first 240 odd numbers.

(6) Find the average of the first 3862 even numbers.

(7) What will be the average of the first 4863 odd numbers?

(8) Find the average of odd numbers from 13 to 1283

(9) Find the average of the first 1877 odd numbers.

(10) Find the average of odd numbers from 13 to 1161


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