Question

a savings account is opened with $15 and increases monthly by 20%. Complete the graph of the geometric sequence by labeling the points

Answer 1

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):

1. Month 0:
\[
A_0 = 15
\]

2. Month 1:
\[
A_1 = 15 \cdot (1.20)^1 = 15 \cdot 1.20 = 18
\]

3. Month 2:
\[
A_2 = 15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.60
\]

4. Month 3:
\[
A_3 = 15 \cdot (1.20)^3 = 15 \cdot 1.728 = 25.92
\]

5. Month 4:
\[
A_4 = 15 \cdot (1.20)^4 = 15 \cdot 2.0736 \approx 31.10
\]

6. Month 5:
\[
A_5 = 15 \cdot (1.20)^5 = 15 \cdot 2.48832 \approx 37.33
\]

7. 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.