Questions: Find three more terms for the following sequence. 0,0.5,0.666..., 0.75,0.8,0.833..., What is the seventh term of the sequence? Choose the correct answer below. A. 0.85714... B. 0.857142... C. 0.8571428... D. 0.8571...

Solution Steps

To find the next terms in the sequence, we need to identify the pattern. The given sequence is: \[ 0, 0.5, 0.\overline{6}, 0.75, 0.8, 0.8\overline{3} \]

We observe that the sequence appears to be increasing and the terms are fractions or repeating decimals. We can convert these decimals to fractions to see if there's a pattern in the denominators or numerators.

1. Convert the given decimals to fractions.
2. Identify the pattern in the sequence of fractions.
3. Use the identified pattern to find the next three terms.
4. Convert the next three terms back to decimals.

The given sequence is: \[ 0, 0.5, 0.\overline{6}, 0.75, 0.8, 0.8\overline{3} \] We convert these decimals to fractions: \[ 0 = \frac{0}{1}, \quad 0.5 = \frac{1}{2}, \quad 0.\overline{6} = \frac{2}{3}, \quad 0.75 = \frac{3}{4}, \quad 0.8 = \frac{4}{5}, \quad 0.8\overline{3} = \frac{5}{6} \]

The fractions can be expressed as: \[ \frac{n}{n+1} \quad \text{for } n = 0, 1, 2, 3, 4, 5 \] Following this pattern, the next three terms correspond to \( n = 6, 7, 8 \): \[ \frac{6}{7}, \quad \frac{7}{8}, \quad \frac{8}{9} \]

Calculating the decimal values of the next three terms: \[ \frac{6}{7} \approx 0.8571, \quad \frac{7}{8} = 0.875, \quad \frac{8}{9} \approx 0.8889 \]

The seventh term of the sequence is the first of the next three terms: \[ 0.8571 \]

Final Answer