In this set of exercises, you will use sequences and their...

In this set of exercises, you will use sequences and their sums to study real-world problems. The parents of a newborn child decide to start saving for the child's college education. At the end of each calendar year, they put $\$ 1500$ into an Educational Savings Account (ESA) that pays $6 \%$ interest compounded annually. What will be the total amount in the account 18 years after they make their initial deposit?

In this set of exercises, you will use sequences and their sums to study real-world problems.
The parents of a newborn child decide to start saving for the child's college education. At the end of each calendar year, they put $\$ 1500$ into an Educational Savings Account (ESA) that pays $6 \%$ interest compounded annually. What will be the total amount in the account
18 years after they make their initial deposit?

Key Concepts

Sequences

Sequences represent ordered lists of numbers that follow a specific rule. They provide the framework for modeling processes where each term builds on the previous ones, making them essential for understanding progression in many real-world applications, including finance.

Series and Their Sums

Series are formed when the terms of a sequence are added together. The concept of summing a series is critical in problems where accumulated values are calculated over time, such as computing the total growth of periodic investments or savings.

Geometric Series

A geometric series is a type of series in which each term is obtained by multiplying the previous term by a constant ratio. This is particularly useful in compound interest calculations, where the interest earned each period is a fixed percentage of the previous balance, leading to exponential growth.

Compound Interest

Compound interest is the process where interest is calculated on the initial principal as well as on the accumulated interest from previous periods. This concept is key in financial computations because it demonstrates how investments or debts grow over time.

Annuities

Annuities involve making or receiving regular payments over a period of time. In finance, calculating the future value of an annuity typically uses the summation of a geometric series, which accounts for each deposit growing compounded over time.

Transcript

00:02 In this scenario, a child's parents puts $1 ,500 into a bank account right after the child is born.

00:08 And this bank account accumulates compound interest to 6%.

00:11 So compound interest means that for any given year, the bank is crediting the interest of 6 % of the amount that is currently, the current balance back into the bank, back into the savings account.

00:27 And that interest can be applied on top of.

00:30 The previous interest.

00:31 So that's what that is what is called a compound interest system.

00:35 So let's write out the first few years and see there's a pattern here.

00:41 So at the end of the first year, we have 1500 times.

00:45 We can convert this into a decimal so it would be easier to multiply and do algebra with.

00:50 So 6 % is equal to 0 .06 just by moving the decimal back from 6 .0 back to spaces to the left.

00:59 And we can rewrite this, we can write it here as 1 plus 0 .06.

01:04 And the purpose of the 1 over here is that it stands for 100%.

01:08 So this means that at the end of the 15, at the end of the first year, we're keeping the same amount.

01:13 We're keeping the original 1500 in the accounts.

01:16 And we're simply adding interest on top of that 0 .06 times 1 ,500.

01:23 So we have for after the end of the first year, 1 ,500, times 1 .06.

01:29 And at the end of the second year, we have the same amount, 1500 times 1 .06...

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