Question

The variance of the numbers 8, 21, 34, 47, \dots, 320, is:

Hint:

For an arithmetic progression, the variance can be calculated using the formula \( \frac{n}{12} d^2 \), where \( n \) is the number of terms and \( d \) is the common difference.

Answer 1

Correct Answer: 8788

The given numbers form an arithmetic progression with first term \( a = 8 \), common difference \( d = 13 \), and the last term \( l = 320 \). The number of terms \( n \) in the sequence is given by: \[ l = a + (n - 1)d \quad \Rightarrow \quad 320 = 8 + (n - 1) \cdot 13, \] \[ 320 = 8 + 13n - 13 \quad \Rightarrow \quad 320 = 13n - 5 \quad \Rightarrow \quad 325 = 13n \quad \Rightarrow \quad n = 25. \] The variance of an arithmetic sequence is given by: \[ \text{Variance} = \frac{n}{12} \cdot d^2. \] Substitute \( n = 25 \) and \( d = 13 \): \[ \text{Variance} = \frac{25}{12} \cdot 13^2 = \frac{25}{12} \cdot 169 = \frac{4225}{12} = 8788. \]