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Question :    Find the average of the first 2841 odd numbers.


Correct Answer  2841

Solution & 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 2841 odd numbers are

1, 3, 5, 7, 9, . . . . 2841 th terms

Calculation of the sum of the first 2841 odd numbers

We can find the sum of the first 2841 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 2841 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 2841 odd number,

n = 2841, a = 1, and d = 2

Thus, sum of the first 2841 odd numbers

S2841 = 2841/2 [2 × 1 + (2841 – 1) 2]

= 2841/2 [2 + 2840 × 2]

= 2841/2 [2 + 5680]

= 2841/2 × 5682

= 2841/2 × 5682 2841

= 2841 × 2841 = 8071281

⇒ The sum of first 2841 odd numbers (Sn) = 8071281

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 2841 odd numbers

= 28412 = 8071281

⇒ The sum of first 2841 odd numbers = 8071281

Calculation of the Average of the first 2841 odd numbers

Formula to find the Average

Average = Sum of given numbers/Number of numbers

Thus, The average of the first 2841 odd numbers

= Sum of first 2841 odd numbers/2841

= 8071281/2841 = 2841

Thus, the average of the first 2841 odd numbers = 2841 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 2841 odd numbers = 2841

Thus, the average of the first 2841 odd numbers = 2841 Answer


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