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. $$\begin{array}{|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous Compounding }} \\ \hline A & {} & {} \\ \hline\end{array}$$ $$ \begin{array}{l}{P=\$ 1000} \\ {r=3 \frac{1}{2} \%} \\ {t=10 \text { years }}\end{array} $$

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.
$$\begin{array}{|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous Compounding }} \\ \hline A & {} & {} \\ \hline\end{array}$$
$$
\begin{array}{l}{P=\$ 1000} \\ {r=3 \frac{1}{2} \%} \\ {t=10 \text { years }}\end{array}
$$

Key Concepts

Compound Interest

Compound interest is the process by which interest is earned on both the original principal and any accumulated interest from previous periods. This results in exponential growth over time, making it a fundamental concept in finance for calculating investment and loan growth.

Periodic Compounding

Periodic compounding involves applying the compound interest formula by dividing the annual interest rate by the number of compounding periods per year. The total amount accumulated is given by the formula A = P(1 + r/n)^(n*t), where each period’s interest is reinvested, thus generating additional interest in subsequent periods.

Continuous Compounding

Continuous compounding is an idealized form of compounding where interest is added an infinite number of times per year, leading to the formula A = P * e^(r*t). This method leverages the natural exponential function, resulting in a slightly higher accumulated amount compared to standard periodic compounding under the same interest rate.

Transcript

0:00 Hello.

00:02 So here we're given that p is equal to $1 ,000.

00:06 R is equal to 3 .5%.

00:08 So that's 7 .5 % or 0 .035 and t is equal to 10 years.

00:14 So we know that the balance after t years is given by, while a, the amount after two years, is equal to p, the principle times 1 plus r over n to the n t.

00:28 Where n is the number of compounding times per year.

00:33 And when we're compounded continuously, well, then we have a is equal to p times the number e raised to the rt power.

00:41 So here, when n is equal to one using a is equal to p times while one plus r, just one plus r to the t, we get the amount a is equal to, well, the principle here which is 1 ,000 times 1 plus 0 .035 so that's 1 .035 so plus 1 plus 0 .035 to the 10 this is going to be equal to $1 ,410 and we get 0 .59 so that's around to 60 cents and when t is equal to or n is equal to two so that's when ends equal to 1 when n is equal to 2 then we have the amount a is equal to well again 1 ,000 times 1 plus 0 .035 over 2 and then to the n times 2 so now to the 20th and that is going to be equal to 1 ,400 $114 .78.

02:03 And we could do when n is equal to four, and when n is equal to 12, and then when n is equal to 365, and when we do n is equal to 365, well, then we get 1 ,000 times 1 plus 0 .035 over 365 to the 1 out 3 ,650, and that would give us 1 ,490.

02:29 So not much more, a little bit more, and four cents.

02:33 So this is when n is equal to 365.

02:37 And if we did compounding continuously, well, then we would have what? we would have a is equal to 1 ,000 times e raised to the 0 .035 times 10, which is going to be approximately equal to 1 ,000.

02:59 $1 ,419 still and about $7.

03:11 Okay, so completing the chart.

03:14 Well, we would have our n and we would have our a, and then when n is one, we get that a is $1 ,410 .60...

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