Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


2 , 10 , 50 , ...


Write your answer using decimals and integers.


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

Answer 1

To find an equation for the sequence \(2, 10, 50, \ldots\), we first examine the pattern of the terms.

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

2. **Finding a Pattern:**
- We can observe the relationship between the terms:
- \(10\) is \(2 \times 5\) (or \(2 \times 5^1\))
- \(50\) is \(10 \times 5\) (or \(2 \times 5^2\))
- It appears the terms are \(2\) multiplied by increasing powers of \(5\).

3. **Defining the General Term:**
- The first term can be expressed as:
- \(a_1 = 2 = 2 \times 5^0\)
- \(a_2 = 10 = 2 \times 5^1\)
- \(a_3 = 50 = 2 \times 5^2\)

This suggests that the \(n\)-th term can be represented as:
\[
a_n = 2 \times 5^{n-1}
\]

Putting it all together, the equation for the \(n\)-th term in the sequence is:

\[
a_n = 2(5)^{n-1}
\]