Given $a_1 = -3$ and $r = 5$, this is a geometric sequence where each term is obtained by multiplying the previous term by the common ratio $r$.

Explicit Rule:

The explicit rule for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Substitute $a_1 = -3$ and $r = 5$: $a_n = -3 \cdot 5^{n-1}$

Recursive Rule:

The recursive rule is: $a_1 = -3$ $a_{n} = 5 \cdot a_{n-1} \quad \text{for } n > 1$

Next Two Terms:

Using the explicit rule $a_n = -3 \cdot 5^{n-1}$:

• $a_2 = -3 \cdot 5^{2-1} = -3 \cdot 5 = -15$
• $a_3 = -3 \cdot 5^{3-1} = -3 \cdot 25 = -75$

Thus, the next two terms are: $a_2 = -15, \quad a_3 = -75$

Find the 15th Term:

Using the explicit rule: $a_{15} = -3 \cdot 5^{15-1} = -3 \cdot 5^{14}$ Calculate $5^{14}$ and multiply by $-3$: $a_{15} = -3 \cdot 6103515625 = -18310546875$

Final Answer:

• Explicit Rule: $a_n = -3 \cdot 5^{n-1}$
• Recursive Rule: $a_1 = -3, \; a_n = 5 \cdot a_{n-1}$
• Next Two Terms: $a_2 = -15, \; a_3 = -75$
• 15th Term: $a_{15} = -18310546875$

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Do you want further explanations or examples? Here are five related questions for exploration:

1. What is the sum of the first 15 terms in this geometric sequence?
2. How would the explicit formula change if $r = -5$?
3. How can you graph this geometric sequence on a coordinate plane?
4. What is the ratio between consecutive terms in this sequence, and why is it constant?
5. How does changing $a_1$ affect the sequence's behavior?

Tip: To quickly find large terms in a geometric sequence, use logarithms to simplify calculations for powers.