Question

Reaching a Financial Goal You need to accumulate $10,000. To do so, you plan to make deposits of $1,400 per year - with the first payment being made a year from today - into a bank account that pays 10% annual interest. Your last deposit will be less than $1,400 if less is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal? Do not round intermediate calculations. Round your answer up to the nearest whole number. year(s) How large will the last deposit be? Do not round intermediate calculations. Round your answer to the nearest cent. $

Answer 1

To reach a financial goal of $10,000 with annual deposits of $1,400 at 10% annual interest, the future value of an annuity formula is used to calculate the number of years and the final deposit. Compound interest significantly enhances savings over time.

Step-by-step explanation:

To reach a financial goal of accumulating $10,000, one must consider the effects of compound interest when making regular deposits. In this specific scenario, a student is making annual deposits of $1,400 into an account that yields a 10% annual interest rate. To find out how many years it will take to reach the $10,000 goal, we need to use the future value of an annuity formula, which is given by

FV = Pmt * (((1 + r)^n - 1) / r)

Where FV is the future value of the annuity, Pmt is the annual payment, r is the annual interest rate, and n is the number of payments. Rearranging the formula to solve for n gives:

n = ln[(FV * r / Pmt) + 1] / ln[1 + r]

By inputting the values ($10,000 for FV, 0.10 for r, and $1,400 for Pmt) into the rearranged formula, we can calculate the number of years required to reach the goal. The last deposit can be calculated by finding the accumulated value just before the last deposit and then determining how much is needed to reach the $10,000 target. The exact amount of the last payment is the difference between $10,000 and the future value after n-1 deposits.

The provided information allows an inference into the power of compound interest, where consistent, early savings and the accrual of interest over time substantially increase the sum of funds.