# Find the average of all odd numbers up to 100: (1) 30, (2) 50, (3) 4, (4) 10

The average is the ratio of the sum of all given observations to the total number of observations.

## Answer: The average of all odd numbers up to 100 is 50, which is option (2).

Let's understand the formula of the average and thereby find the solution to the given problem.

**Explanation:**

The list of odd numbers up to 100: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .........., 99.

They form an arithmetic progression with a common difference 2.

Sum = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 .................+ 99

a = 1, d = 2 , L = 99

The sum of the first n terms of an arithmetic sequence when the nth term is known is: \(S_{n}\) = (n/2) × [a + L].

⇒ \(S_{100}\)_{ }= (100/2) × [ 1 + 99 ]

⇒ \(S_{100}\)_{ }= (100/2) × 100

⇒ \(S_{100}\) = 50 × 100 = 5,000

The average is given by the formula: Average = sum of the observations/number of observations.

Average = sum of odd numbers up to 100 / 100 = 5000/100 = 50.

### Thus, the average of all odd numbers up to 100 is 50, which is option (2).

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