Arithmetic and geometric sequences, and compound interest

What you are looking at

An arithmetic sequence adds the same common difference $d$ at every step, so $u_n = u_1 + (n-1)d$ and its bars rise or fall in a straight line. A geometric sequence multiplies by the same common ratio $r$, so $u_n = u_1 r^{n-1}$. The visualiser draws both from the same first term, green for arithmetic and blue for geometric, and the working panel gives the $n$th term and the sum $S_n$ of each for the current values.

Compound interest is a geometric sequence in disguise. With an annual rate of $r\%$ compounded $k$ times a year for $n$ years, the future value is $FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}$. The blue step graph jumps once per compounding period. As $k$ goes from annual to daily the steps get smaller and the final value creeps up, but it never passes the dashed continuous limit $PV\,e^{rn/100}$.

Try this

1. 1Press Add terms and watch the bars appear one at a time. With the starting values ($u_1 = 2$, $d = 1.5$, $r = 1.3$) the arithmetic sequence leads at first, but the geometric one overtakes it at the 8th term.
2. 2Drag $r$ below 1, say to $0.8$, and the blue bars shrink towards zero. Then try a negative $r$ and watch them alternate above and below the axis.
3. 3Switch to Compound interest and press Compound more often. $k$ steps through 1, 2, 4, 12 and 365: a present value of 1000 at 5% for 10 years grows to 1628.89 compounded annually and 1648.66 daily, against a limit of 1648.72.
4. 4Press Hide formulas. The working is replaced by a Predict first question (and in Compound interest mode the dashed limit curve is hidden), so you can commit to an answer before checking.

Mistakes students make

• Writing $u_n = u_1 + nd$. The first term has had no differences added, so the $n$th term has $n - 1$ of them: $u_n = u_1 + (n-1)d$, and in the same way $u_n = u_1 r^{n-1}$.
• Using the annual rate and the number of years when interest is compounded monthly. The rate per period is $\dfrac{r}{100k}$ and the number of periods is $kn$, so monthly compounding over 10 years means 120 periods.
• Assuming that compounding more often makes the value grow without bound. It does increase, but towards the finite limit $PV\,e^{rn/100}$: in the example above, going from monthly to daily adds only 1.65.

Where it is taught

• IB Maths AA SL 1.2–1.4 · HL 1.2–1.4 (arithmetic sequences, geometric sequences, financial applications)
• Edexcel A-Level Maths Pure Year 2 · Arithmetic and geometric sequences and series
• Greek Lykeio (Α΄ Λυκείου) · Άλγεβρα, chapter 5: Πρόοδοι (arithmetic and geometric progressions)