Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

Question

–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.