Average
MCQs Math

Question:   ( 1 of 10 )  Find the average of the first 2571 odd numbers.

You selected   1285.5

Correct Answer  2571

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 2571 odd numbers are

1, 3, 5, 7, 9, . . . . 2571 th terms

Calculation of the sum of the first 2571 odd numbers

We can find the sum of the first 2571 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 2571 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

S_n = ⁿ/₂ [2a + (n – 1) d]

Where, n = number of terms, a = first term, and d = common difference

In the series of first 2571 odd number,

n = 2571, a = 1, and d = 2

Thus, sum of the first 2571 odd numbers

S₂₅₇₁ = ²⁵⁷¹/₂ [2 × 1 + (2571 – 1) 2]

= ²⁵⁷¹/₂ [2 + 2570 × 2]

= ²⁵⁷¹/₂ [2 + 5140]

= ²⁵⁷¹/₂ × 5142

= ²⁵⁷¹/₂ × 5142 2571

= 2571 × 2571 = 6610041

⇒ The sum of first 2571 odd numbers (S_n) = 6610041

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 2571 odd numbers

= 2571² = 6610041

⇒ The sum of first 2571 odd numbers = 6610041

Calculation of the Average of the first 2571 odd numbers

Formula to find the Average

Average = ^{Sum of given numbers}/_{Number of numbers}

Thus, The average of the first 2571 odd numbers

= ^{Sum of first 2571 odd numbers}/₂₅₇₁

= ^6610041/₂₅₇₁ = 2571

Thus, the average of the first 2571 odd numbers = 2571 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 2571 odd numbers = 2571

Thus, the average of the first 2571 odd numbers = 2571 Answer