13/14 13. Sequences, Series, and Mathematical Modeling

A structured guide to sequences, series, and mathematical modeling, including arithmetic and geometric patterns, finite and infinite sums, exponential growth and decay, polynomial models, and model evaluation.

Sequences and Their Representations

A is an ordered list of numbers, and each number is a term identified by its position. The notation $a_n$ means the term in position $n$ . For example, the $5,8,11,14,\ldots$ has $a_1=5$ , $a_2=8$ , and so on.

An gives a term directly from its index. If $a_n=3n+2$ , then

a_{10}=3(10)+2=32.

A uses earlier terms and must include an initial condition. The rules

a_1=5,\qquad a_n=a_{n-1}+3\quad(n\ge 2)

generate the same . Recursive formulas naturally describe repeated steps, while explicit formulas are usually more efficient for finding a distant term.

Takeaway: Use the index to identify a term, an explicit rule for direct access, and a recursive rule when each step depends on what came before.

Arithmetic Sequences

An has a constant difference between consecutive terms. If the is $d$ , then

a_n=a_1+(n-1)d.

For $7,12,17,22,\ldots$ , the is $d=5$ , so

a_n=7+(n-1)(5)=5n+2.

The twentieth term is

a_{20}=7+19(5)=102.

The difference may be negative, producing a decreasing , or zero, producing a constant . If two terms are known, their positions can determine the difference. For example, if $a_4=18$ and $a_{10}=42$ , then

42-18=(10-4)d,

so $d=4$ . Substituting into $a_4=a_1+3d$ gives $a_1=6$ , and therefore $a_n=6+4(n-1)=4n+2$ .

Takeaway: Equal additive changes indicate an arithmetic pattern; find the before using the .

Geometric Sequences

A has a constant ratio between consecutive nonzero terms. If the is $r$ , then

a_n=a_1r^{n-1}.

For $3,12,48,192,\ldots$ , each term is multiplied by $4$ , so $r=4$ and

a_n=3(4)^{n-1}.

The eighth term is

a_8=3(4)^7=49{,}152.

When $0<r<1$ , the terms decrease toward zero. When $r>1$ , their magnitudes generally increase. A negative ratio causes the signs to alternate.

To distinguish the two major patterns, check consecutive changes. The $4,9,14,19,\ldots$ has constant differences and is arithmetic. The $4,12,36,108,\ldots$ has a constant ratio and is geometric.

Takeaway: Equal additive changes indicate arithmetic behavior; equal multiplicative factors indicate geometric behavior.

Finite and Sums

A is formed by adding the terms of a . Sigma notation gives a compact representation:

\sum_{k=1}^{n}a_k=a_1+a_2+\cdots+a_n.

For an , the sum of the first $n$ terms is

S_n=\frac{n}{2}(a_1+a_n)

or, after substituting the formula for the final term,

S_n=\frac{n}{2}\bigl[2a_1+(n-1)d\bigr].

For the first 20 terms of $7,12,17,\ldots$ , first calculate $a_{20}=102$ , then

S_{20}=\frac{20}{2}(7+102)=1090.

For a finite geometric with $r\ne 1$ , use

S_n=a_1\frac{1-r^n}{1-r}.

For $2,6,18,54,\ldots$ , the first five terms sum to

S_5=2\frac{1-3^5}{1-3}=242.

When $r=1$ , every term equals $a_1$ , so $S_n=na_1$ .

Takeaway: Arithmetic sums depend on the first and last terms, while geometric sums depend on the initial term, ratio, and number of terms.

Infinite Geometric

An infinite geometric has a finite sum only when the satisfies $|r|<1$ . In that case,

S_\infty=\frac{a_1}{1-r}.

For example,

10+5+2.5+1.25+\cdots

has $a_1=10$ and $r=\frac{1}{2}$ , so

S_\infty=\frac{10}{1-\frac{1}{2}}=20.

If $|r|\ge 1$ , the infinite does not converge to a finite sum. A ratio greater than or equal to $1$ prevents the terms from approaching zero; a sufficiently negative ratio can cause alternating terms whose magnitudes do not shrink.

Takeaway: Before applying the infinite-sum formula, verify the essential condition $|r|<1$ .

Recognizing and Choosing Models

A connects a real situation to mathematical quantities. A practical modeling process is:

Identify the quantities and their units.
Define variables and determine whether inputs are discrete or continuous.
Examine the data for differences, ratios, or finite-difference patterns.
Choose a model type and determine its parameters.
Check the model against known values.
Interpret the result in context.
State limitations and identify whether the prediction is or .

Use a when the input is discrete, such as payment number or generation number. A function is often more suitable when the input can vary continuously, such as time or distance.

Equal changes in output for equal input intervals suggest linear or arithmetic behavior. Equal percentage changes or multiplication by a constant factor suggest exponential or geometric behavior. Constant higher-order finite differences can suggest a .

For example, $100,110,121,133.1,\ldots$ has $1.10$ , so an appropriate model is

a_n=100(1.10)^{n-1}.

In contrast, $100,110,120,130,\ldots$ has $10$ , so an appropriate model is

a_n=100+10(n-1).

Takeaway: Select a model from the structure of the data, then test whether its predictions make sense in the real context.

Exponential Growth, Decay, and Interest

An has the form

A(t)=A_0b^t,

where $A_0$ is the initial amount and $b$ is the growth factor. For a percentage growth rate $p$ , use $b=1+p$ . For a percentage decay rate $p$ , use $b=1-p$ , with the rate written as a decimal.

A population of 12,000 growing by $3\%$ per year can be modeled by

P(t)=12{,}000(1.03)^t.

After five years,

P(5)=12{,}000(1.03)^5\approx13{,}911.

For depreciation, a machine worth $\$18{,}000$ that loses $12\%$ of its value each year has model

V(t)=18{,}000(0.88)^t.

After four years, its predicted value is approximately $\$10{,}816$ .

For compound interest, with principal $P$ , annual rate $r$ , $n$ compounding periods per year, and time $t$ in years,

A(t)=P\left(1+\frac{r}{n}\right)^{nt}.

Continuous compounding uses

A(t)=Pe^{rt}.

Takeaway: Convert percentage rates into multiplication factors before building an .

Polynomial Models and Finite Differences

A has the form

P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.

Its degree is the greatest exponent with a nonzero coefficient. Polynomial models can describe curved relationships such as height, area, cost, revenue, and population over a limited interval.

Consider the profit model

P(x)=-0.5x^2+40x-300.

Because the coefficient of $x^2$ is negative, the graph opens downward. Its maximum occurs at

x=-\frac{b}{2a}=-\frac{40}{2(-0.5)}=40.

The corresponding profit is

P(40)=-0.5(40)^2+40(40)-300=500.

Thus, within the range where the model is reasonable, it predicts a maximum profit of $\$500$ at 40 units.

Finite differences provide a clue about degree. Constant first differences suggest a linear model, constant second differences suggest a quadratic model, and constant third differences suggest a cubic model. This evidence should be checked against the context because polynomial predictions may become unrealistic far beyond the data interval.

Takeaway: Use finite differences to suggest a polynomial degree, but restrict conclusions to a domain supported by the situation.

Evaluating Models and Their Limits

Model evaluation requires more than matching a pattern. Check the following questions:

Are the inputs discrete or continuous?
Are consecutive outputs separated by a constant difference?
Do consecutive outputs have a constant ratio?
Do finite differences become constant?
Does the model have a sensible domain and range?
Is the estimate within the observed interval, or does it go beyond the data?
Does the predicted result have a reasonable interpretation and unit?

estimates within the available data range. predicts beyond it and should be treated cautiously because the observed pattern may not continue.

A model can fit known values and still be inappropriate if it violates the context. For instance, a geometric pattern may predict continued population growth, but the assumption of a constant growth rate may eventually fail. A polynomial may fit a short interval well while producing unrealistic values outside that interval.

Final takeaway: A strong model combines an appropriate algebraic pattern with a meaningful context, a justified domain, and an honest statement of limitations.