Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

–2 , –10 , –50 , ...


Write your answer using decimals and integers.


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

Answer 1

To derive an equation for the sequence -2, -10, -50, ..., we first need to observe the pattern of the terms.

1. **Identify the terms:**
- \( a_1 = -2 \)
- \( a_2 = -10 \)
- \( a_3 = -50 \)

2. **Finding the ratio of consecutive terms:**
- \(\frac{a_2}{a_1} = \frac{-10}{-2} = 5\)
- \(\frac{a_3}{a_2} = \frac{-50}{-10} = 5\)

This indicates that the terms are generated by multiplying the previous term by 5.

3. **Identifying the relationship with n:**
- Each term can be expressed as a function of its position. It appears that each term can be represented as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where \( a_1 = -2 \) (the first term) and \( r = 5 \) (the common ratio).

4. **Substituting the values:**
- So, substituting the values we have:
\[
a_n = -2 \cdot 5^{(n-1)}
\]

In conclusion, the equation for the nth term in the sequence is:
\[
a_n = -2 \cdot 5^{n-1}
\]

So, in the specified format:
\[
a_n = -2(5)^{n-1}
\]