To illustrate the growth of a savings account that starts with $15 and increases by 20% each month, we can represent this with a geometric sequence.
The initial amount is:
- \( a_0 = 15 \)
The growth factor is:
- \( r = 1 + 0.20 = 1.20 \)
The formula for the \( n \)-th term of a geometric sequence is:
- \( a_n = a_0 \cdot r^n \)
Using this formula, we can calculate the first few terms of the sequence.
-
Month 0 (initial amount):
- \( a_0 = 15 \)
-
Month 1:
- \( a_1 = 15 \cdot (1.20)^1 = 15 \cdot 1.20 = 18.00 \)
-
Month 2:
- \( a_2 = 15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.60 \)
-
Month 3:
- \( a_3 = 15 \cdot (1.20)^3 = 15 \cdot 1.728 = 25.92 \)
-
Month 4:
- \( a_4 = 15 \cdot (1.20)^4 = 15 \cdot 2.0736 = 31.10 \)
-
Month 5:
- \( a_5 = 15 \cdot (1.20)^5 = 15 \cdot 2.48832 = 37.32 \)
To summarize, here are the points to label on the graph:
- Month 0: \( (0, 15) \)
- Month 1: \( (1, 18.00) \)
- Month 2: \( (2, 21.60) \)
- Month 3: \( (3, 25.92) \)
- Month 4: \( (4, 31.10) \)
- Month 5: \( (5, 37.32) \)
You can plot these points on a graph where the x-axis represents the months and the y-axis represents the account balance in dollars. The growth will create an exponential curve, reflecting the nature of geometric sequences.
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