Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

Question

1 , 5 , 25 , ...

Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To find the equation that describes the sequence \(1, 5, 25, \ldots\), we can analyze the terms.

1. The first term (\(n = 1\)) is \(1\).
2. The second term (\(n = 2\)) is \(5\).
3. The third term (\(n = 3\)) is \(25\).

We can notice that these terms can also be expressed as powers of \(5\):

• \(a_1 = 5^0 = 1\)
• \(a_2 = 5^1 = 5\)
• \(a_3 = 5^2 = 25\)

In general, it appears that the \(n\)-th term of the sequence can be expressed as: \[ a_n = 5^{n - 1} \]

So the final equation for the \(n\)-th term of the sequence is: \[ a_n = 5^{n - 1} \]

Answer 2

answer this with decimals and intergers

Answer 3

Certainly! To express the equation in terms of integers and decimals, we can retain the structure of powers but revise the notation accordingly. The equation for the sequence using integers and decimals can be written as:

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

However, if you want to explicitly show decimals in the equation but maintain the correct values, we can represent it as:

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

Alternatively, if you are looking for a more explicit decimal representation for each term, we could also express the equation as:

\[ a_n = 5^{(n - 1)} = (5.0)^{(n - 1)} \]

However, the standard representation \(a_n = 5^{(n - 1)}\) is already in a clear integer exponent format with the base as decimal. If you need any further modifications, please let me know!