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. $$\begin{array}{|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous Compounding }} \\ \hline A & {} & {} \\ \hline\end{array}$$ $$ \begin{array}{l}{P=\$ 2500} \\ {r=6 \%} \\ {t=20 \text { years }}\end{array} $$

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.
$$\begin{array}{|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous Compounding }} \\ \hline A & {} & {} \\ \hline\end{array}$$
$$
\begin{array}{l}{P=\$ 2500} \\ {r=6 \%} \\ {t=20 \text { years }}\end{array}
$$

Key Concepts

Compound Interest Formula

This formula determines the future value of an investment by taking into account the effect of interest added to the principal over multiple periods. It is expressed as A = P(1 + r/n)^(n*t), where A is the amount, P is the principal, r is the annual interest rate (in decimal form), t is the time in years, and n is the number of compounding periods per year. The formula illustrates how not only the initial principal but also the accumulated interest contributes to the overall growth of the investment.

Periodic Compounding

Periodic compounding refers to the process of calculating and adding interest to the original investment balance at regular intervals during the year. The number of times that interest is compounded (such as annually, semi-annually, quarterly, monthly, or daily) can affect the overall amount accrued. More frequent compounding periods result in interest being calculated on a continuously increasing balance, thereby yielding a higher final amount compared to less frequent compounding.

Continuous Compounding

Continuous compounding is an idealized form of compounding where interest is added an infinite number of times per year. Instead of using the periodic compound interest formula, continuous compounding is calculated using the formula A = P * e^(r*t), where e is the base of the natural logarithm. This represents the mathematical limit of the compound interest model as the number of compounding periods per year becomes infinitely large.

Exponential Growth in Finance

Exponential growth describes the process by which a quantity increases at a rate proportional to its current value, resulting in the increasingly rapid escalation of the amount over time. In the context of compound interest, exponential growth illustrates how repeated application of interest over time leads to a dramatic increase in the investment's value. This concept is fundamental in understanding how even small differences in the interest rate or compounding frequency can lead to considerable differences in the future value of an investment.

Transcript

0:00 Hello.

00:01 So here we're given that our principal p is equal to $2 ,500.

00:06 The interest rate r is equal to 6%.

00:09 So 0 .06 and our time t is equal to 20 years.

00:13 So we know that the balance that we have after t years is going to be equal to a is equal to the principal p times 1 plus r over n raised to the nt power, where n here is the number of compounding times per year.

00:34 And if we have continuous compounding, well, then we have that the amount is equal to just perch, where p times e raised to the rt power.

00:45 So here, when n is equal to one, well, then we have a is equal to just p times one plus r to the t.

00:52 So that's equal.

00:53 This is again, when n is equal to one, we have 2 ,500 times, times 1 plus 0 .06 raised to the 20th, which is going to give us $8 ,017 .84.

01:15 So in the, i'm thinking our table, right, in the first, in the first, when n equals one, we get $8 ,017 .84.

01:24 Then when n is equal to two, well, then we have, we have, have a is equal to p times one plus r over two raised to the two t so that's going to be 2 500 times one plus 0 .06 over two raised to the 40th um which is going to give us um 8 ,100 155 dollars and nine cents.

01:59 Okay so there is what our goals at our table when n is equal to two then when n is equal to four well now we're going to use again a is equal to p times now we have one plus r over four and then raised to the four t so that is going to be we get a is equal to well again 2500 is our p times one plus the r which is 0 .06 now over n so over four and then to the four times t so four times 20 is going to give us 80 so to the 80th power and that is going to work out to 8 ,20026.

02:51 So there what goes in our table when n is equal to four and then when we have n is equal to 12, well, now we have a is equal to p times 1 plus r over 12 to the 12 t.

03:06 So that's going to be 2 ,500 times 1 plus 0 .06 over 12.

03:16 Now raised to the 12t, so 12 times 20, which is going to be to the 240th power...

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