Many real-world problems can be modeled with sequences and series. A sequence describes values step by step.
A series adds those values.
1. Sequence model
A sequence is an ordered list:
\[
a_0,\ a_1,\ a_2,\ldots
\]
A sequence can be written explicitly:
\[
a_n=a_0r^n
\]
or recursively:
\[
a_{n+1}=ra_n+b
\]
2. Series model
A series adds sequence terms:
\[
S_N=a_1+a_2+\cdots+a_N
\]
For a geometric series:
\[
S_N=a\frac{1-r^N}{1-r}
\qquad (r\ne1)
\]
3. Bouncing ball model
Suppose a ball is dropped from height \(h_0\), and after each bounce it reaches a fraction \(r\) of the previous height.
The bounce heights form a geometric sequence:
\[
h_n=h_0r^n
\]
The total distance after \(N\) bounces is the first drop plus twice each rebound height:
\[
D_N=h_0+2(h_0r+h_0r^2+\cdots+h_0r^N)
\]
\[
D_N=h_0+2h_0\frac{r(1-r^N)}{1-r}
\]
If \(0\le r\lt1\), the infinite total distance is:
\[
D_\infty=h_0+2h_0\frac{r}{1-r}
\]
4. Loan amortization model
A loan balance changes by adding interest and subtracting the payment:
\[
B_{n+1}=B_n(1+i)-P
\]
Here \(i\) is the interest rate per payment period, \(P\) is the payment, and \(B_n\) is the balance after \(n\) payments.
If the payment is calculated automatically, the standard amortizing payment is:
\[
P=\frac{B_0i}{1-(1+i)^{-N}}
\]
If \(i=0\), then:
\[
P=\frac{B_0}{N}
\]
5. Savings and annuity model
Savings with regular deposits can be modeled by:
\[
A_{n+1}=A_n(1+i)+d
\]
Here \(A_n\) is the balance, \(i\) is the interest rate per period, and \(d\) is the deposit each period.
The closed form is:
\[
A_N=A_0(1+i)^N+d\frac{(1+i)^N-1}{i}
\]
If \(i=0\), then:
\[
A_N=A_0+dN
\]
6. Population growth model
A population may grow by a percentage and also receive a fixed addition:
\[
P_{n+1}=P_n(1+r)+b
\]
The value \(r\) is the growth rate per period, and \(b\) is the fixed added amount.
This model is useful for population, customers, subscribers, or inventory.
7. Decay and depreciation model
When an amount loses the same percentage each period, it follows a geometric decay sequence:
\[
A_n=A_0(1-d)^n
\]
Here \(d\) is the decay rate. This can model radioactive-style decay, cooling patterns, depreciation, or repeated loss.
8. Choosing the correct template
| Scenario |
Model type |
Main formula |
| Bouncing ball |
Geometric series |
\(D_N=h_0+2h_0\dfrac{r(1-r^N)}{1-r}\) |
| Loan |
Recursive balance |
\(B_{n+1}=B_n(1+i)-P\) |
| Savings |
Annuity recurrence |
\(A_{n+1}=A_n(1+i)+d\) |
| Population |
Growth recurrence |
\(P_{n+1}=P_n(1+r)+b\) |
| Decay / depreciation |
Geometric sequence |
\(A_n=A_0(1-d)^n\) |
9. Graph interpretation
The calculator graph shows two pieces of information:
- The bars show the change, payment, deposit, loss, or rebound amount for each step.
- The curve shows the main modeled value, such as balance, population, remaining amount, or total distance.
If the curve levels off, the sequence may be approaching a limiting value.
If it keeps rising or falling, the model may be growing or decreasing without a fixed bound over the selected time range.
10. Common mistakes
- Do not confuse a sequence term with a cumulative series total.
- For bouncing balls, remember that rebound heights are traveled twice: up and down.
- For loans, convert the annual interest rate to the payment-period rate.
- For savings, know whether deposits happen once per month, once per year, or at another frequency.
- For population models, distinguish percentage growth from fixed additions.
- For decay, use \(1-d\), not \(d\), as the remaining fraction each period.
