What is the average of odd numbers from 1 to 40?

Question:What is the average of odd numbers from 1 to 40?

Choices:

• 20

• 21

• 31

• 41

Correct Answer: 20

Explanation:

To find the average of odd numbers between 1 and 40, let's follow a systematic approach.

First, we need to identify the odd numbers in the given range. Odd numbers between 1 and 40 are:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39

These are consecutive odd numbers forming an arithmetic series where each number differs from the previous by a common difference of 2.

Step-by-Step Calculation:

1. First term of the series, $a = 1$

2. Last term of the series, $l = 39$

3. The series has a common difference, $d = 2$

The number of terms, $n$, in the sequence can be found using the arithmetic sequence formula:

$n = \frac{l - a}{d} + 1$

Substitute the values:

$n = \frac{39 - 1}{2} + 1 = 20$

Now, to find the average of these 20 terms, we first need to know the sum of these terms. The sum, $S_n$, of the first $n$ terms in an arithmetic sequence is given by:

$S_n = \frac{n}{2} \times (a + l)$

Substitute the values:

$S_{20} = \frac{20}{2} \times (1 + 39) = 10 \times 40 = 400$

Now, we find the average by dividing the sum by the number of terms:

$\text{Average} = \frac{S_{20}}{20} = \frac{400}{20} = 20$

Therefore, the average of the odd numbers from 1 to 40 is 20.