Question

Write an equation to describe the sequence below.Use n to represent the position of a term in the sequence,where n = 1 for the first term.
1/7 , -5/21 , 25/63 ,
Write your answer using proper fractions, improper fractions, and integers.

an = __ ( __ )^n-1

Answer 1

To describe the sequence \( \frac{1}{7}, -\frac{5}{21}, \frac{25}{63} \) using the formula \( a_n = a( r )^{n-1} \) where \( a \) is the first term and \( r \) is the common ratio, let's first analyze the terms of the sequence.

1. Identify the first term:
- \( a_1 = \frac{1}{7} \)

2. Identify the second term:
- \( a_2 = -\frac{5}{21} \)

3. Identify the third term:
- \( a_3 = \frac{25}{63} \)

Next, let's check how to find the common ratio \( r \):
- The common ratio can be found by dividing the second term by the first term:
\[
r = \frac{a_2}{a_1} = \frac{-\frac{5}{21}}{\frac{1}{7}} = -\frac{5}{21} \times \frac{7}{1} = -\frac{5 \times 7}{21 \times 1} = -\frac{35}{21} = -\frac{5}{3}
\]

Now, let's check the ratio between the third and second terms:
\[
r = \frac{a_3}{a_2} = \frac{\frac{25}{63}}{-\frac{5}{21}} = \frac{25}{63} \times -\frac{21}{5} = -\frac{25 \times 21}{63 \times 5} = -\frac{525}{315} = -\frac{5}{3}
\]

Both calculations give \( r = -\frac{5}{3} \).

Now we can plug in the values into the sequence formula:
- Since \( a = \frac{1}{7} \) and \( r = -\frac{5}{3} \):

So, the equation that describes the sequence is:
\[
a_n = \frac{1}{7} \left( -\frac{5}{3} \right)^{n-1}
\]

This is the final formula for the given sequence.