Compound Interest In Exercises 89-92, complete the table by...

Compound Interest In Exercises $89-92,$ complete the table by determining the balance $A$ for $P$ dollars invested at rate $r$ for $t$ years and compounded $n$ times per year. $$\begin{array}{|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous Compounding }} \\ \hline A & {} & {} \\ \hline\end{array}$$ $$ \begin{array}{l}{P=\$ 4000} \\ {r=4 \%} \\ {t=15 \text { years }}\end{array} $$

Compound Interest In Exercises $89-92,$ complete the table by determining the balance $A$ for $P$ dollars invested at rate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous Compounding }} \\ \hline A & {} & {} \\ \hline\end{array}$$
$$
\begin{array}{l}{P=\$ 4000} \\ {r=4 \%} \\ {t=15 \text { years }}\end{array}
$$

Key Concepts

Compound Interest

Compound interest is the process of earning interest on both the initial principal and the accumulated interest from previous periods. It is fundamental to finance and investment, showing how investments can grow over time by reinvesting earned interest.

Discrete Compounding

Discrete compounding involves calculating interest at specific, regular intervals such as annually, semiannually, quarterly, monthly, or daily. The formula A = P(1 + r/n)^(n*t) is used to compute the future value, where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is time in years.

Continuous Compounding

Continuous compounding is the concept where compounding occurs at every possible instant, rather than at set intervals. This is modeled by the formula A = Pe^(rt), using the mathematical constant e. It represents the limit of the compound interest formula as the number of compounding periods per year increases indefinitely.

Transcript

0:00 Hello.

00:01 So we know that if we invest p dollars at a rate of r percent for t years compounded n times per year, then the amount we have after t years compounded n times per year is given by a is equal to p times a quantity 1 plus r over n raised to the nt power.

00:21 And if we're compounded continuously, well then we have the amount is just equal to the same principle p but then times e.

00:29 Number e raised to the rt power.

00:32 So here, let's consider we have our principle of $4 ,000, that our interest rate is 4%, so 0 .04, and our time t is equal to 15 years.

00:44 So when n is equal to 1, well, then what do we have? we have a is equal to, again, p times 1 plus r for n to the nt.

00:56 So that is going to be 4 ,000, and then times, well, one plus 0 .04 over 1 is just times 1 .04.

01:08 And then raised to the, well, here it's t times 1.

01:11 So just the 15th power.

01:13 And that is going to work out to be $7 ,203.

01:21 And 77 cents.

01:23 So there we'll go in our table for when n is equal to 1.

01:29 Now it's considering when n is equal to 2.

01:30 So when n is equal to two, what do we have? well, we have our principal p, so 4 ,000 times 1 plus r over n.

01:39 That's going to be 1 plus 0 .04 over 2, which is going to be times 1 .02, and then raised to the nt.

01:47 So raised to the 2 times 15 or raised to the 30th power, and that's going to give us $7 ,245.

01:57 And $44.

01:58 So that would go in our table for when n is equal to 2.

02:03 Now, let's consider when n is equal to 4.

02:06 So when n is equal to 4, we have the amount a is equal to the principal p, so $4 ,000, then times 1 plus r over n.

02:16 So it's 1 plus 0 .04 divided by 4, which is going to give us, i believe, 1 .01, and then raise that to the 4 times 15, or raise that to the 4 .4.

02:26 60th power and that's going to give us $7 ,266 .78.

02:36 So that's going to go on a table for when n is equal to 4...

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