Answer :
Let's analyze the given sequence and derive a recursive formula for it.
The sequence provided is:
[tex]\[ 5, 25, 125, 625, 3125, 15625, \ldots \][/tex]
We notice the following:
1. The first term [tex]\( a_1 \)[/tex] is given as [tex]\( 5 \)[/tex].
2. To find the recursive formula, let's observe the pattern in the consecutive terms:
[tex]\[
\begin{aligned}
a_2 &= 25 \\
a_3 &= 125 \\
a_4 &= 625 \\
a_5 &= 3125 \\
a_6 &= 15625 \\
\end{aligned}
\][/tex]
3. We can determine how each term relates to the previous one by calculating the ratios:
[tex]\[
\begin{aligned}
\frac{a_2}{a_1} &= \frac{25}{5} = 5 \\
\frac{a_3}{a_2} &= \frac{125}{25} = 5 \\
\frac{a_4}{a_3} &= \frac{625}{125} = 5 \\
\frac{a_5}{a_4} &= \frac{3125}{625} = 5 \\
\frac{a_6}{a_5} &= \frac{15625}{3125} = 5 \\
\end{aligned}
\][/tex]
4. From this pattern, we observe that each term is obtained by multiplying the previous term by [tex]\( 5 \)[/tex].
5. Therefore, a recursive formula to express the [tex]\( n \)[/tex]-th term in terms of the [tex]\((n-1)\)[/tex]-th term can be written as:
[tex]\[
a_n = a_{n-1} \times 5 \text{ for } n > 1
\][/tex]
Let’s formally state the recursive formula:
[tex]\[
\begin{cases}
a_1 = 5 \\
a_n = a_{n-1} \times 5 & \text{for } n > 1
\end{cases}
\][/tex]
This is the recursive formula that defines the sequence correctly based on the given terms.
The sequence provided is:
[tex]\[ 5, 25, 125, 625, 3125, 15625, \ldots \][/tex]
We notice the following:
1. The first term [tex]\( a_1 \)[/tex] is given as [tex]\( 5 \)[/tex].
2. To find the recursive formula, let's observe the pattern in the consecutive terms:
[tex]\[
\begin{aligned}
a_2 &= 25 \\
a_3 &= 125 \\
a_4 &= 625 \\
a_5 &= 3125 \\
a_6 &= 15625 \\
\end{aligned}
\][/tex]
3. We can determine how each term relates to the previous one by calculating the ratios:
[tex]\[
\begin{aligned}
\frac{a_2}{a_1} &= \frac{25}{5} = 5 \\
\frac{a_3}{a_2} &= \frac{125}{25} = 5 \\
\frac{a_4}{a_3} &= \frac{625}{125} = 5 \\
\frac{a_5}{a_4} &= \frac{3125}{625} = 5 \\
\frac{a_6}{a_5} &= \frac{15625}{3125} = 5 \\
\end{aligned}
\][/tex]
4. From this pattern, we observe that each term is obtained by multiplying the previous term by [tex]\( 5 \)[/tex].
5. Therefore, a recursive formula to express the [tex]\( n \)[/tex]-th term in terms of the [tex]\((n-1)\)[/tex]-th term can be written as:
[tex]\[
a_n = a_{n-1} \times 5 \text{ for } n > 1
\][/tex]
Let’s formally state the recursive formula:
[tex]\[
\begin{cases}
a_1 = 5 \\
a_n = a_{n-1} \times 5 & \text{for } n > 1
\end{cases}
\][/tex]
This is the recursive formula that defines the sequence correctly based on the given terms.
