For the following exercises, use the compound interest...

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
An account is opened with an initial deposit of $\$ 6,500$ and earns 3.6$\%$ interest compounded semi-annually. What will the account be worth in 20 years?

Key Concepts

Exponential Growth

Exponential growth describes the process where a quantity increases over time at a rate proportional to its current value. In compound interest, this is observed as the investment grows more rapidly because each compounding period calculates interest on an ever-expanding base.

Compound Interest Formula

This formula calculates the future value of an investment or loan by applying interest not only on the initial principal but also on the accumulated interest from previous periods. The formula A(t) = P(1 + r/n)^(n*t) is fundamental in finance for modeling compounded growth over time.

Principal

The principal is the initial sum of money invested or deposited. It forms the basis for the compound interest calculation, as all interest computations build upon this original amount.

Interest Rate

The interest rate represents the percentage at which the principal grows within a given time period. In the context of compound interest, the annual rate is divided by the number of compounding periods, influencing how much interest is accrued in each period.

Compounding Frequency

Compounding frequency indicates how often interest is calculated and added to the principal. More frequent compounding results in interest being computed on an increasingly larger amount, which can dramatically affect the total accumulation over time.

Transcript

00:01 All right, so use the compound interest formula to solve this question.

00:07 So an account is open with initial deposit of $6 ,500, and it earns 3 .6 % interest compounded semi -annually.

00:16 And what will the account be worth that for 20 years? okay, so let's go to our whiteboard.

00:21 And we have, oops, let me, this is from the last question.

00:26 We already have the equation, so that's good.

00:28 Let me just erase partially.

00:31 It's like part of it.

00:34 P and then just, if not i can just redraw it, but i just want to keep it.

00:47 We just make a little bit nicer.

00:49 And all right.

00:51 So we have a to the t or a of t is equal to p times 1 plus r of n to the n over t.

00:57 And to solve the question, you really have to know what all these variables mean.

01:01 And once you know that, it's very, very easy.

01:05 So we have the answer right here.

01:07 The final answer after a certain amount of years a of t right this is a standard function you plug in something for t t goes right there and then you just solve it out so pretty straightforward p is the starting value the principal amount it's the amount you had in the beginning and i think that was 6500 i have to double check but we'll do that all in a sec let me just tell you what the rest of the variables mean first one is just one you know just there to um make the uh final rate uh 1 point something it's going to be like some rate.

01:38 And the rate's going to be calculated by r over n.

01:41 So r is the actual percentage rate.

01:44 And then n is the amount of times it's compounded every year.

01:48 So it's semi -annually...

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