Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.
–2 , –6/5, –18/25, ...
Write your answer using proper fractions, improper fractions, and integers.

an = _(_)^ n^ -^ 1

Answer 1

To derive the equation for the sequence \(-2, -\frac{6}{5}, -\frac{18}{25}, \ldots\), we first need to identify any patterns.

1. **Identify the terms:**
- \(a_1 = -2\)
- \(a_2 = -\frac{6}{5}\)
- \(a_3 = -\frac{18}{25}\)

2. **Express the terms in fractional form:**
- The first term is \(-2 = -\frac{10}{5}\).
- The second term remains \(-\frac{6}{5}\).
- The third term can be expressed as \(-\frac{18}{25}\).

3. **Look for patterns in the numerators and denominators:**
- Numerators: \(-10, -6, -18\)
- These can be represented as:
- \(10 = 2 \times 5\)
- \(6 = 6\)
- \(18 = 6 \times 3\)
- Denominators: \(5, 5, 25\)
- The denominators seem to follow the powers of \(5\).

4. **Finding a rule:**
After observing the numerators and denominators, notice that:
- The numerators follow the sequence:
- The pattern here can be \(3 \times 2^{n-1}\) (where \(n\) represents position).
- The denominators seem to be \(5^{n-1}\) (as \(5^1 = 5\), \(5^2 = 25\), etc.).

5. **Combining these** into a formula for the \(n^{th}\) term:
\[
a_n = -\frac{3 \cdot 2^{n-1}}{5^{n-1}}
\]

This formula represents each term of the sequence. Therefore, the equation describing the sequence is:

\[
a_n = -\frac{3 \cdot 2^{n-1}}{5^{n-1}}
\]

You can also write it slightly differently using properties of exponents:

\[
a_n = -3 \cdot \left(\frac{2}{5}\right)^{n-1}
\]

So, depending on the preferred format, you can use either version to describe the sequence.