To graph the savings account which starts with $15 and increases by 20% each month, we can represent this situation as a geometric sequence.
The first term (initial amount) is \( a = 15 \), and the common ratio \( r \) is \( 1 + 0.20 = 1.20 \).
The formula for the \( n \)-th term of a geometric sequence is:
\[ a_n = a \cdot r^{(n-1)} \]
Calculating the first few terms:
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Month 1 (n=1): \[ a_1 = 15 \cdot (1.20)^{0} = 15 \cdot 1 = 15 \]
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Month 2 (n=2): \[ a_2 = 15 \cdot (1.20)^{1} = 15 \cdot 1.20 = 18 \]
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Month 3 (n=3): \[ a_3 = 15 \cdot (1.20)^{2} = 15 \cdot 1.44 = 21.60 \]
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Month 4 (n=4): \[ a_4 = 15 \cdot (1.20)^{3} = 15 \cdot 1.728 = 25.92 \]
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Month 5 (n=5): \[ a_5 = 15 \cdot (1.20)^{4} = 15 \cdot 2.0736 = 31.10 \]
Now we can label points on the graph based on these values:
- Month 1: \( (1, 15) \)
- Month 2: \( (2, 18) \)
- Month 3: \( (3, 21.60) \)
- Month 4: \( (4, 25.92) \)
- Month 5: \( (5, 31.10) \)
Now, plot these points on a graph with the x-axis representing the month (n) and the y-axis representing the amount in the savings account (a_n). The points will show the exponential growth of the savings account over the months. Each point represents how much money is in the account at the end of each month.
Feel free to ask if you need further assistance with specific graphing techniques or tools!
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