The given recursive formula for the geometric sequence is:

$a_n = 5(a_{n-1}), \quad a_1 = 4$

To express this sequence using function notation, we need to find a general expression for $a_n$ in terms of $n$.

Step 1: Analyze the recursive formula

The recursive formula tells us that each term is 5 times the previous term. So, starting from the first term:

• $a_1 = 4$
• $a_2 = 5a_1 = 5 \times 4 = 20$
• $a_3 = 5a_2 = 5 \times 20 = 100$
• $a_4 = 5a_3 = 5 \times 100 = 500$
• And so on...

Step 2: Generalize the formula

Notice that each term is a multiple of 5 raised to the power of $n-1$, starting from the first term $a_1 = 4$. This suggests that the general formula for the $n$-th term can be written as:

$a_n = 4 \times 5^{n-1}$

Final expression:

Thus, the explicit formula for the geometric sequence in function notation is:

$a_n = 4 \times 5^{n-1}$

This is the general expression for the $n$-th term of the sequence.

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Would you like further details or have any questions on this? Here are 5 related questions for you:

1. How does the general formula $a_n = 4 \times 5^{n-1}$ help you find any term in the sequence?
2. How would the formula change if the common ratio were different (e.g., 3 instead of 5)?
3. What is the sum of the first 5 terms in this geometric sequence?
4. How does the recursive definition relate to the explicit formula in geometric sequences?
5. How would you approach solving for a specific term if $a_n = 4 \times 5^{n-1}$ was modified by a different starting value?

Tip: In geometric sequences, the general form for the $n$-th term is often $a_n = a_1 \times r^{n-1}$, where $r$ is the common ratio.