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

To find the variance of the sequence of numbers 8, 21, 34, 47, ..., 320, we start by identifying it as an arithmetic sequence. The first term \(a=8\), the common difference \(d=21-8=13\), and the last term \(l=320\).

Step 1: Determine the Number of Terms (n) 

The nth term of an arithmetic sequence is given by:
\( a_n = a + (n-1)d \)

Setting \( a_n = 320 \):
\(320 = 8 + (n-1) \times 13\)
\(320 - 8 = (n-1) \times 13\)
\(312 = (n-1) \times 13\)
\(n-1 = \frac{312}{13} = 24\)
\(n = 25\)

Step 2: Calculate the Mean (\(\bar{x}\))

The mean is:
\(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{1}{25}(8 + 21 + 34 + \cdots + 320)\)

The sum of an arithmetic sequence is calculated by:
\(S_n = \frac{n}{2}(a + l)\)

Thus:
\(S_{25} = \frac{25}{2}(8 + 320) = \frac{25}{2} \times 328 = 4100\)

The mean is:
\(\bar{x} = \frac{4100}{25} = 164\)

Step 3: Calculate the Variance (\(\sigma^2\))

Variance is defined as:
\(\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 \)

For an arithmetic sequence, the variance formula simplifies, and we can calculate using:
\(\sigma^2 = \frac{1}{12}(n^2-1)d^2 \)

Substituting in the values:
\(\sigma^2 = \frac{1}{12}(25^2-1)\times 13^2\)
\(\sigma^2 = \frac{1}{12}(624)\times 169\)
\(\sigma^2 = \frac{1}{12}(105456)\)
\(\sigma^2 = \frac{8788}{1}\)

The calculated variance is 8788.

Answer 2

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 \( d = 13 \): \[ \text{Variance} = \frac{25}{12} \cdot 13^2 = \frac{25}{12} \cdot 169 = \frac{4225}{12} = 8788. \]