An investor deposits 10,000 in an account that earns 3.5 %...

An investor deposits $\$ 10,000$ in an account that earns $3.5 \%$ interest compounded quarterly. The balance in the account after $n$ quarters is given by $$1-\log _{a} a\left(1+\frac{a_{a} a_{2}}{4}\right)=n-1,23 \ldots$$ (a) Write the first eight terms of the sequence. (b) Find the balance in the account after 10 years by computing the 40 th term of the sequence. (c) Is the balance after 20 years twice the balance after 10 years? Explain.

An investor deposits $\$ 10,000$ in an account that earns $3.5 \%$ interest compounded quarterly. The balance in the account after $n$ quarters is given by $$1-\log _{a} a\left(1+\frac{a_{a} a_{2}}{4}\right)=n-1,23 \ldots$$
(a) Write the first eight terms of the sequence.
(b) Find the balance in the account after 10 years by computing the 40 th term of the sequence.
(c) Is the balance after 20 years twice the balance after
10 years? Explain.

Key Concepts

Compound Interest

Compound interest is the process by which an investment grows over time as the interest earned on the principal earns additional interest. This concept involves calculating interest not just on the original principal but also on the accumulated interest from previous periods, making it a cornerstone of financial mathematics and investment growth.

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed constant, known as the common ratio. In financial contexts, the account balance growing under compound interest forms a geometric sequence, with the common ratio reflecting the periodic growth factor due to interest.

Time Period Conversion

Time period conversion involves translating time spans into the appropriate units when working with compound interest. For instance, when interest is compounded quarterly, the annual interest rate must be appropriately divided and the number of years converted to the number of quarters, ensuring the calculations accurately reflect the compounding frequency.

Exponential Growth

Exponential growth describes processes that increase at a rate proportional to their current value. In the context of compound interest, the repeated application of the growth factor leads to exponential increases in the account balance over time, illustrating how small periodic interest rates can lead to significant growth over long periods.

Transcript

00:01 In this problem of here series, we have said that an investor he is depositing, an investor deposits $10 ,000.

00:11 So this is $10 ,000 in an account and he is earning, this is at the rate of 3 .5 % compound interest quarterly.

00:21 So this is 3 .5 % per quarter.

00:25 This is a quarterly.

00:26 Quarterly interest, that is 3 .5%.

00:29 The balance in the account after n quarters is given by the balance in the account after n quarters here is the number of quarters is given why this is 10 ,000 multiplied with here 1 plus this is 0 .035 divided with 4 to the whole power n.

00:52 And now we have where n is the number of quarter 1, 2, 3 and so on.

00:59 And now we have to write the first eight terms of the sequence first.

01:04 So we have to write the first eight terms.

01:07 So here put here, n is equal to 1.

01:09 So this would be here 10 ,000 1 plus 0 .035 divide with 4 to the power 1.

01:18 So this is the first term.

01:20 So this would be here by simplifying we get here.

01:24 A1 equals to this is 10 ,000 here.

01:30 087 this is 10 ,087 .5 this is 0 .50 dollar this is the amount of first quarter and then second quarter simplifying this term here 1 plus 0 .035 to the power 4 to whole here 2 so this is the amount in second quarter that is equal to here dollar $10 ,175 .55.

02:10 And similarly, a3 is dollar simplifying, this is the, we have to simplify this term, 1 plus 0 .035 divide with 4 to whole power 3 that is here.

02:25 10 ,264 .80.

02:29 So this is 10 ,264.

02:33 Point eight zero dollar and now a four is here ten thousand this is three and fifty four point six two dollar and fifth that is a five is here ten thousand this is ten thousand four hundred and forty five point two dollar and now here is a six equals to this is ten thousand this is dollar ten thousand this is dollar ten 5 ,536 .62 and now 8 .7 that is here is 10 ,600, this is $10 ,628 .81.

03:21 And now the 8th is, this is $10 ,727 .27.

03:28 This is the $10 ,7 .27.

03:32 This is the answer of first part.

03:36 This is the here amount after number of quarters.

03:43 So a 1 represent the amount after first quarter a2 represent the amount after second quarter and now 8 represent the amount after 8th quarter...

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