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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

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Using the compound interest formula to solve for the principal

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 r = 0.06. Interest is compounded twice a year, so k = 2.

We want to find the initial investment, P , needed so that the value of the account will be worth $40,000 in 18 years. Substitute the given values into the compound interest formula, and solve for P .

A ( t ) = P ( 1 + r n ) n t Use the compound interest formula . 40,000 = P ( 1 + 0.06 2 ) 2 ( 18 ) Substitute using given values  A r ,   n , and  t . 40,000 = P ( 1.03 ) 36 Simplify . 40,000 ( 1.03 ) 36 = P Isolate  P . P $ 13 , 801 Divide and round to the nearest dollar .

Lily will need to invest $13,801 to have $40,000 in 18 years.

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Refer to [link] . To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?

$13,693

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Evaluating functions with base e

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 ) = ( 1 + 1 n ) n Value
Annually ( 1 + 1 1 ) 1 $2
Semiannually ( 1 + 1 2 ) 2 $2.25
Quarterly ( 1 + 1 4 ) 4 $2.441406
Monthly ( 1 + 1 12 ) 12 $2.613035
Daily ( 1 + 1 365 ) 365 $2.714567
Hourly ( 1 + 1 8766 ) 8766 $2.718127
Once per minute ( 1 + 1 525960 ) 525960 $2.718279
Once per second ( 1 + 1 31557600 ) 31557600 $2.718282

These values appear to be approaching a limit as n increases without bound. In fact, as n gets larger and larger, the expression ( 1 + 1 n ) n approaches a number used so frequently in mathematics that it has its own name: the letter e . 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 number e

The letter e represents the irrational number

( 1 + 1 n ) n , as n increases without bound

The letter e is used as a base for many real-world exponential models. To work with base e , we use the approximation, e 2.718282. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

Using a calculator to find powers of e

Calculate e 3.14 . Round to five decimal places.

On a calculator, press the button labeled [ e x ] . The window shows [ e ^ (   ] . Type 3.14 and then close parenthesis, [ ) ] . Press [ENTER]. Rounding to 5 decimal places, e 3.14 23.10387. 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 e .

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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