Math Problem Statement

An initial investment of
$ 1 , 000 at
12 % interest is compounded weekly (use
52 weeks in a year). What will the investment be worth in
30 years? Round your answer to the nearest cent.

Solution

To solve this problem, we'll use the formula for compound interest:

A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}

where:

  • AA is the amount of money accumulated after nn years, including interest.
  • PP is the principal amount (the initial investment).
  • rr is the annual interest rate (in decimal form).
  • nn is the number of times that interest is compounded per year.
  • tt is the number of years the money is invested for.

Given:

  • P=1000P = 1000 dollars
  • r=12%=0.12r = 12\% = 0.12
  • n=52n = 52 (since the interest is compounded weekly)
  • t=30t = 30 years

Let's calculate AA:

A=1000(1+0.1252)52×30A = 1000 \left(1 + \frac{0.12}{52}\right)^{52 \times 30}

Now, we'll calculate the value.

The investment will be worth $36,446.76 after 30 years, when compounded weekly at a 12% annual interest rate.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

  1. How would the final amount change if the interest were compounded daily instead of weekly?
  2. What would be the investment worth after 20 years instead of 30?
  3. How does the compound interest formula differ from simple interest?
  4. How would the amount change if the interest rate were 10% instead of 12%?
  5. How does increasing the compounding frequency affect the final investment amount?

Tip: The more frequently interest is compounded, the greater the final amount will be due to the effect of earning interest on interest more often.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Exponential Growth

Formulas

Compound Interest Formula

Theorems

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Suitable Grade Level

High School