Average
MCQs Math


Question:     Find the average of the first 2277 odd numbers.


Correct Answer  2277

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 2277 odd numbers are

1, 3, 5, 7, 9, . . . . 2277 th terms

Calculation of the sum of the first 2277 odd numbers

We can find the sum of the first 2277 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 2277 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 2277 odd number,

n = 2277, a = 1, and d = 2

Thus, sum of the first 2277 odd numbers

S2277 = 2277/2 [2 × 1 + (2277 – 1) 2]

= 2277/2 [2 + 2276 × 2]

= 2277/2 [2 + 4552]

= 2277/2 × 4554

= 2277/2 × 4554 2277

= 2277 × 2277 = 5184729

⇒ The sum of first 2277 odd numbers (Sn) = 5184729

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 2277 odd numbers

= 22772 = 5184729

⇒ The sum of first 2277 odd numbers = 5184729

Calculation of the Average of the first 2277 odd numbers

Formula to find the Average

Average = Sum of given numbers/Number of numbers

Thus, The average of the first 2277 odd numbers

= Sum of first 2277 odd numbers/2277

= 5184729/2277 = 2277

Thus, the average of the first 2277 odd numbers = 2277 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 2277 odd numbers = 2277

Thus, the average of the first 2277 odd numbers = 2277 Answer


Similar Questions

(1) Find the average of the first 604 odd numbers.

(2) Find the average of even numbers from 6 to 82

(3) What will be the average of the first 4629 odd numbers?

(4) Find the average of the first 1461 odd numbers.

(5) Find the average of odd numbers from 9 to 1191

(6) Find the average of odd numbers from 5 to 311

(7) Find the average of even numbers from 12 to 1380

(8) What is the average of the first 1101 even numbers?

(9) Find the average of the first 2796 odd numbers.

(10) Find the average of the first 2190 odd numbers.


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