# 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: Option (2) 50, is the average of all odd numbers up to 100.

Let's understand the formula of the average and thereby find the solution to a 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 , T_{n} = 99 = l

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 / 10 = (1 + 3 + 5+ 7 + 9+ .........+ 99) / 100 = 5000/100 = 50.