Compound Interest 5C9A9A

1. **State the problem:** We start with an initial deposit of $3000, and each year an additional $3000 is deposited. The account earns 6% interest compounded annually. We want to find the balance after the seventh deposit. 2. **Formula used:** This is a problem involving the future value of an annuity with compound interest. The formula for the future value $FV$ after $n$ deposits is: $$FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r) + P$$ where: - $P$ is the annual deposit ($3000$), - $r$ is the annual interest rate (6% or 0.06), - $n$ is the number of deposits minus 1 (since the initial deposit is separate). However, since the initial deposit is made at the start and then 6 more deposits are made annually, the total number of deposits is 7. Alternatively, we can consider the initial deposit grows for 7 years, and each subsequent deposit grows for fewer years. 3. **Calculate the future value of the initial deposit:** $$FV_{initial} = 3000 \times (1 + 0.06)^7 = 3000 \times 1.503630 = 4510.89$$ 4. **Calculate the future value of the 6 subsequent deposits:** Each deposit is made at the end of each year, so the first additional deposit grows for 6 years, the second for 5 years, ..., the last for 1 year. The sum of these is an annuity: $$FV_{annuity} = 3000 \times \frac{(1 + 0.06)^6 - 1}{0.06} \times (1 + 0.06)$$ Calculate: $$(1 + 0.06)^6 = 1.418519$$ So: $$FV_{annuity} = 3000 \times \frac{1.418519 - 1}{0.06} \times 1.06 = 3000 \times \frac{0.418519}{0.06} \times 1.06$$ Calculate the fraction: $$\frac{0.418519}{0.06} = 6.97532$$ Multiply: $$3000 \times 6.97532 \times 1.06 = 3000 \times 7.39384 = 22181.52$$ 5. **Total balance after 7 deposits:** $$FV = FV_{initial} + FV_{annuity} = 4510.89 + 22181.52 = 26692.41$$ 6. **Check the problem statement:** The problem states "after the seventh deposit," so the last deposit has just been made and has not earned interest yet. We must subtract the interest on the last deposit. The last deposit has 0 years to grow, so we subtract $3000$ from the total to get the balance immediately after the seventh deposit: $$26692.41 - 3000 = 23692.41$$ This does not match the options, so let's use the standard annuity formula for deposits made at the beginning of each period (annuity due): $$FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$$ Here, $n=7$, $P=3000$, $r=0.06$: Calculate: $$(1 + 0.06)^7 = 1.50363$$ $$\frac{1.50363 - 1}{0.06} = \frac{0.50363}{0.06} = 8.39383$$ Multiply: $$3000 \times 8.39383 \times 1.06 = 3000 \times 8.89746 = 26692.38$$ This is the total balance after the seventh deposit and interest for that year. 7. **Re-examining the problem:** The problem likely assumes the first deposit is made at time zero and the last deposit is made at the end of the seventh year, so the last deposit has no interest yet. Therefore, the balance after the seventh deposit (just after deposit, before interest) is the future value of 6 deposits plus the last deposit: Calculate the future value of 6 deposits: $$(1 + 0.06)^6 = 1.41852$$ $$\frac{1.41852 - 1}{0.06} = 6.9753$$ $$3000 \times 6.9753 \times 1.06 = 22181.5$$ Add the last deposit: $$22181.5 + 3000 = 25181.5$$ Rounded to nearest dollar: $25182$ 8. **Final answer:** The balance after the seventh deposit is $25182$. **Answer choice:** $25,182$.