To represent the growth of a savings account that starts with $15 and increases by 20% monthly, we define the geometric sequence.
Each month's amount can be calculated using the formula for a geometric sequence:
\[ A_n = A_0 \cdot r^n \]
Where:
- \( A_0 \) is the initial amount ($15)
- \( r \) is the growth factor (1 + growth rate), which is \( 1 + 0.20 = 1.20 \)
- \( n \) is the number of months
Now, we can compute the amounts for the first few months (let’s calculate for 6 months):
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Month 0: \[ A_0 = 15 \]
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Month 1: \[ A_1 = 15 \cdot (1.20)^1 = 15 \cdot 1.20 = 18 \]
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Month 2: \[ A_2 = 15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.60 \]
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Month 3: \[ A_3 = 15 \cdot (1.20)^3 = 15 \cdot 1.728 = 25.92 \]
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Month 4: \[ A_4 = 15 \cdot (1.20)^4 = 15 \cdot 2.0736 \approx 31.10 \]
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Month 5: \[ A_5 = 15 \cdot (1.20)^5 = 15 \cdot 2.48832 \approx 37.33 \]
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Month 6: \[ A_6 = 15 \cdot (1.20)^6 = 15 \cdot 2.985984 \approx 44.79 \]
Here are the amounts for each month, which you can plot on a graph:
- Month 0: \( (0, 15) \)
- Month 1: \( (1, 18) \)
- Month 2: \( (2, 21.60) \)
- Month 3: \( (3, 25.92) \)
- Month 4: \( (4, 31.10) \)
- Month 5: \( (5, 37.33) \)
- Month 6: \( (6, 44.79) \)
You can label these points on your graph for clearer visualization. The x-axis represents the number of months, and the y-axis represents the amount in the savings account. The points will show an upward exponential curve, illustrating the compound growth of the account.