Answer :
We start by identifying the first term of the sequence:
[tex]$$a_1 = -5.$$[/tex]
Next, notice that each term in the sequence is obtained by multiplying the previous term by [tex]$5$[/tex]. This pattern can be verified as follows:
- The second term is
[tex]$$a_2 = 5 \cdot a_1 = 5 \cdot (-5) = -25.$$[/tex]
- The third term is
[tex]$$a_3 = 5 \cdot a_2 = 5 \cdot (-25) = -125.$$[/tex]
- The fourth term is
[tex]$$a_4 = 5 \cdot a_3 = 5 \cdot (-125) = -625.$$[/tex]
- The fifth term is
[tex]$$a_5 = 5 \cdot a_4 = 5 \cdot (-625) = -3125.$$[/tex]
This confirms that the pattern is consistent. Therefore, the recursive notation for the sequence is given by:
[tex]$$
\begin{cases}
a_1 = -5, \\
a_n = 5\,a_{n-1} \quad \text{for } n > 1.
\end{cases}
$$[/tex]
[tex]$$a_1 = -5.$$[/tex]
Next, notice that each term in the sequence is obtained by multiplying the previous term by [tex]$5$[/tex]. This pattern can be verified as follows:
- The second term is
[tex]$$a_2 = 5 \cdot a_1 = 5 \cdot (-5) = -25.$$[/tex]
- The third term is
[tex]$$a_3 = 5 \cdot a_2 = 5 \cdot (-25) = -125.$$[/tex]
- The fourth term is
[tex]$$a_4 = 5 \cdot a_3 = 5 \cdot (-125) = -625.$$[/tex]
- The fifth term is
[tex]$$a_5 = 5 \cdot a_4 = 5 \cdot (-625) = -3125.$$[/tex]
This confirms that the pattern is consistent. Therefore, the recursive notation for the sequence is given by:
[tex]$$
\begin{cases}
a_1 = -5, \\
a_n = 5\,a_{n-1} \quad \text{for } n > 1.
\end{cases}
$$[/tex]
