To derive the explicit formula for the sequence $a_n$ given the recursive relationship:

$a_n = a_{n-1} \cdot 5 \quad \text{and} \quad a_1 = 2,$

we can rewrite it using the properties of geometric sequences.

Step 1: Recognizing a Geometric Sequence

The recursive formula shows that each term is 5 times the previous term. Thus, this is a geometric sequence with:

• Initial term $a_1 = 2$,
• Common ratio $r = 5$.

Step 2: General Formula for a Geometric Sequence

The explicit formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}.$

Step 3: Substitute the Known Values

Here, $a_1 = 2$ and $r = 5$. Substituting these values into the formula gives: $a_n = 2 \cdot 5^{n-1}.$

Final Answer:

The explicit formula for the sequence is: $a_n = 2 \cdot 5^{n-1}.$

Let me know if you’d like further explanation!

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Related Questions:

1. What is the recursive formula for this sequence, and how does it differ from the explicit formula?
2. How can you calculate the 10th term using the explicit formula?
3. What are the first five terms of this sequence?
4. How do you find the sum of the first $n$ terms of a geometric sequence like this?
5. How does the formula change if the common ratio $r$ is negative?

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Tip:

For geometric sequences, always identify the initial term and common ratio first to easily derive both recursive and explicit formulas.