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An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?
about $3,644,675.88
A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?
The nominal interest rate is 6%, so Interest is compounded twice a year, so
We want to find the initial investment, needed so that the value of the account will be worth $40,000 in years. Substitute the given values into the compound interest formula, and solve for
Lily will need to invest $13,801 to have $40,000 in 18 years.
Refer to [link] . To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?
$13,693
As we saw earlier, the amount earned on an account increases as the compounding frequency increases. [link] shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in [link] .
| Frequency | Value | |
|---|---|---|
| Annually | $2 | |
| Semiannually | $2.25 | |
| Quarterly | $2.441406 | |
| Monthly | $2.613035 | |
| Daily | $2.714567 | |
| Hourly | $2.718127 | |
| Once per minute | $2.718279 | |
| Once per second | $2.718282 |
These values appear to be approaching a limit as increases without bound. In fact, as gets larger and larger, the expression approaches a number used so frequently in mathematics that it has its own name: the letter This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.
The letter e represents the irrational number
The letter e is used as a base for many real-world exponential models. To work with base e , we use the approximation, The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.
Calculate Round to five decimal places.
On a calculator, press the button labeled The window shows Type and then close parenthesis, Press [ENTER]. Rounding to decimal places, Caution: Many scientific calculators have an “Exp” button, which is used to enter numbers in scientific notation. It is not used to find powers of
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