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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 $\text{\hspace{0.17em}}r=\mathrm{0.06.}\text{\hspace{0.17em}}$ Interest is compounded twice a year, so $\text{\hspace{0.17em}}k=2.$
We want to find the initial investment, $\text{\hspace{0.17em}}P,$ needed so that the value of the account will be worth $40,000 in $\text{\hspace{0.17em}}18\text{\hspace{0.17em}}$ years. Substitute the given values into the compound interest formula, and solve for $\text{\hspace{0.17em}}P.$
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 | $A(t)={\left(1+\frac{1}{n}\right)}^{n}$ | Value |
---|---|---|
Annually | ${\left(1+\frac{1}{1}\right)}^{1}$ | $2 |
Semiannually | ${\left(1+\frac{1}{2}\right)}^{2}$ | $2.25 |
Quarterly | ${\left(1+\frac{1}{4}\right)}^{4}$ | $2.441406 |
Monthly | ${\left(1+\frac{1}{12}\right)}^{12}$ | $2.613035 |
Daily | ${\left(1+\frac{1}{365}\right)}^{365}$ | $2.714567 |
Hourly | ${\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}$ | $2.718127 |
Once per minute | ${\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}$ | $2.718279 |
Once per second | ${\left(1+\frac{1}{31557600}\right)}^{31557600}$ | $2.718282 |
These values appear to be approaching a limit as $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ increases without bound. In fact, as $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ gets larger and larger, the expression $\text{\hspace{0.17em}}{\left(1+\frac{1}{n}\right)}^{n}\text{\hspace{0.17em}}$ approaches a number used so frequently in mathematics that it has its own name: the letter $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}e\approx \mathrm{2.718282.}\text{\hspace{0.17em}}$ The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.
Calculate $\text{\hspace{0.17em}}{e}^{3.14}.\text{\hspace{0.17em}}$ Round to five decimal places.
On a calculator, press the button labeled $\text{\hspace{0.17em}}\left[{e}^{x}\right].\text{\hspace{0.17em}}$ The window shows $\text{\hspace{0.17em}}\left[e^(\text{}\right].\text{\hspace{0.17em}}$ Type $\text{\hspace{0.17em}}3.14\text{\hspace{0.17em}}$ and then close parenthesis, $\text{\hspace{0.17em}}[)].\text{\hspace{0.17em}}$ Press [ENTER]. Rounding to $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ decimal places, $\text{\hspace{0.17em}}{e}^{3.14}\approx \mathrm{23.10387.}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}e.$
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