Interest as a Sequence

Money over time is the most useful place these two strands meet. Simple interest is an arithmetic sequence wearing a financial hat; compound interest is a geometric one. Once you see that, the formulas stop being things to memorise.

Simple interest is arithmetic growth

Interest calculated only on the original principal adds the same amount every period:

$A=P(1+rt)$

Deposit $\text{textdollar}1000$ at 5% simple interest and the balances are $1050,1100,1150,1200,\dots$ — a common difference of $\text{textdollar}50$, which is an arithmetic sequence with $t_1=1050$ and $d=50$. Plot it and you get points on a straight line, because a constant difference is linear growth. Three descriptions, one behaviour:

Language	The same fact
Financial	Interest on the principal only
Sequence	Arithmetic, common difference $Pr$
Function	Linear, slope $Pr$

Compound interest is geometric growth

Interest calculated on the accumulated balance multiplies instead:

$A=P(1+i{)}^n$

The same $\text{textdollar}1000$ at 5% compounded annually gives $1050,1102.50,1157.63,\dots$ — a common ratio of 1.05. Geometric sequence, exponential function, curve rather than line.

At small $n$ the two are nearly indistinguishable, which is exactly why compounding is easy to underestimate. Over 30 years, simple interest returns $\text{textdollar}2500$ and compound returns about $\text{textdollar}4322$. Same rate, same deposit; the difference is entirely in what the interest is calculated on.

Where the technology belongs

Some questions have no clean algebraic answer. “What monthly payment clears a $\text{textdollar}12,000$ loan in four years at 6.9%?” cannot be rearranged pleasantly by hand, and rearranging it is not the skill being taught.

Use a TVM solver — on a graphing calculator, in a spreadsheet, or one of the standard online tools — and treat it as five quantities where knowing four gives the fifth:

Symbol	What it means
$N$	Number of compounding periods
$I\%$	Annual interest rate, as a percentage
$PV$	Present value — what it is worth now
$PMT$	The regular payment
$FV$	Future value — what it is worth at the end

Two conventions save most of the errors: money coming to you is positive and money leaving you is negative, and $N$ counts compounding periods rather than years. Get either backwards and the answer is confidently wrong.

Always sanity-check the machine

Before accepting a payment figure, multiply it out: 48 payments of $\text{textdollar}287$ is about $\text{textdollar}13,800$ on a $\text{textdollar}12,000$ loan, so roughly $\text{textdollar}1,800$ of interest over four years at 6.9% — plausible. A tool that says $\text{textdollar}87$ a month has been given the rate as 0.069% or the term in months where it wanted years, and the arithmetic check catches it in five seconds.

Your Financial Future is where you use all of this on decisions somebody is actually facing, and Money Over Time has the annuity formulas the solver is doing for you.

Curriculum connection

C3.1

make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology (e.g., use a spreadsheet or graphing calculator to make simple interest calculations, determine first differences in the amounts over time, and graph amount versus time) Sample problem: Describe an investment that could be represented by the function $f(x)=500(1+0.05x)$.

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C3.4

determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, $n$, using the compound interest formula in the form $A=P(1+i{)}^n$ [or $FV=PV(1+i{)}^n$]; describe strategies (e.g., guessing and checking; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems

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