Average
MCQs Math


Question:     Find the average of the first 3512 odd numbers.


Correct Answer  3512

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 3512 odd numbers are

1, 3, 5, 7, 9, . . . . 3512 th terms

Calculation of the sum of the first 3512 odd numbers

We can find the sum of the first 3512 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 3512 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 3512 odd number,

n = 3512, a = 1, and d = 2

Thus, sum of the first 3512 odd numbers

S3512 = 3512/2 [2 × 1 + (3512 – 1) 2]

= 3512/2 [2 + 3511 × 2]

= 3512/2 [2 + 7022]

= 3512/2 × 7024

= 3512/2 × 7024 3512

= 3512 × 3512 = 12334144

⇒ The sum of first 3512 odd numbers (Sn) = 12334144

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 3512 odd numbers

= 35122 = 12334144

⇒ The sum of first 3512 odd numbers = 12334144

Calculation of the Average of the first 3512 odd numbers

Formula to find the Average

Average = Sum of given numbers/Number of numbers

Thus, The average of the first 3512 odd numbers

= Sum of first 3512 odd numbers/3512

= 12334144/3512 = 3512

Thus, the average of the first 3512 odd numbers = 3512 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 3512 odd numbers = 3512

Thus, the average of the first 3512 odd numbers = 3512 Answer


Similar Questions

(1) Find the average of the first 3500 odd numbers.

(2) Find the average of odd numbers from 15 to 1591

(3) Find the average of the first 4069 even numbers.

(4) What is the average of the first 445 even numbers?

(5) Find the average of even numbers from 10 to 1442

(6) Find the average of odd numbers from 11 to 1003

(7) Find the average of the first 1393 odd numbers.

(8) What is the average of the first 235 even numbers?

(9) Find the average of odd numbers from 3 to 271

(10) Find the average of odd numbers from 7 to 1479


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