https://file.aiquickdraw.com/mathbot/file/1734379456164tpr7janc.jpg

To solve this problem, we use the formula for the general term \( a_n \) of a geometric sequence:

\[
a_n = a_1 \cdot r^{n-1}
\]

### Given:
- \( a_1 = 5 \) (the first term)
- \( r = -3 \) (the common ratio)

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### Step 1: Write the formula for \( a_n \):
Substitute \( a_1 = 5 \) and \( r = -3 \) into the formula:

\[
a_n = 5 \cdot (-3)^{n-1}
\]

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### Step 2: Find \( a_5 \) (the 5th term):
To find \( a_5 \), set \( n = 5 \):

\[
a_5 = 5 \cdot (-3)^{5-1}
\]
Simplify the exponent:

\[
a_5 = 5 \cdot (-3)^4
\]

Now calculate \( (-3)^4 \):
\[
(-3)^4 = 81
\]

Thus:
\[
a_5 = 5 \cdot 81 = 405
\]

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### Final Answers:
1. General formula: \( a_n = 5 \cdot (-3)^{n-1} \)
2. \( a_5 = 405 \)

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Let me know if you would like more explanation or further clarifications!

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### Related Questions:
1. What is the 6th term (\( a_6 \)) of this sequence?
2. How does the sign of the common ratio \( r \) affect the sequence?
3. What happens if \( r \) is positive instead of negative?
4. Derive the sum of the first 5 terms of this geometric sequence.
5. What is the \( n \)-th term if \( r = -2 \) and \( a_1 = 4 \)?

### Tip:
In geometric sequences, alternating signs occur if the common ratio \( r \) is negative.

Find a_n and a_5 for the following geometric sequence where a₁ = 5 and r = -3.

Learn how to determine the general formula and the 5th term for a geometric sequence where the first term (a₁) is 5 and the common ratio (r) is -3. Step-by-step solution provided with key formulas.

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