Geometric Sequence A94626

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, $a_1 = 5000$. 4. **Find the common ratio $r$:** Calculate $r$ by dividing the balance at year 2 by the balance at year 1: $$r = \frac{a_2}{a_1} = \frac{5100}{5000} = 1.02$$ 5. **Verify the ratio for other years:** Check year 3: $$\frac{a_3}{a_2} = \frac{5202}{5100} = 1.02$$ Similarly for year 4 and 5, the ratio remains $1.02$, confirming the sequence is geometric. 6. **Write the formula:** $$a_n = 5000 \times (1.02)^{n-1}$$ 7. **Interpretation:** This means the balance grows by 2% each year, starting from $5000$. **Final answer:** $$\boxed{a_n = 5000 \times (1.02)^{n-1}}$$