Geometric Sequence A5F458

1. **State the problem:** We are given a sequence of balances over 5 years and need to find the formula for the geometric sequence that models this data. 2. **Recall the formula for a geometric sequence:** $$a_n = a_1 \times r^{n-1}$$ where $a_n$ is the balance at year $n$, $a_1$ is the initial balance, and $r$ is the common ratio. 3. **Identify the initial term:** From the data, $a_1 = 3000$. 4. **Find the common ratio $r$:** Use the ratio of the second term to the first term: $$r = \frac{a_2}{a_1} = \frac{3150}{3000} = 1.05$$ 5. **Verify the ratio with other terms:** $$\frac{a_3}{a_2} = \frac{3307.50}{3150} = 1.05$$ $$\frac{a_4}{a_3} = \frac{3472.88}{3307.50} \approx 1.05$$ $$\frac{a_5}{a_4} = \frac{3646.52}{3472.88} \approx 1.05$$ All ratios are consistent, confirming $r=1.05$. 6. **Write the formula:** $$a_n = 3000 \times 1.05^{n-1}$$ 7. **Interpretation:** This means the balance grows by 5% each year, starting from $3000$ at year 1. **Final answer:** $$\boxed{a_n = 3000 \times 1.05^{n-1}}$$