4.2 Differences and patterns

A list of values rarely comes with its formula. Looking at the changes between terms is a useful first move: a fixed first difference suggests a line; a fixed second difference suggests a quadratic; a fixed ratio suggests repeated multiplication. This is useful for pattern questions, tables of measurements, and deciding which model to try.

This is foundational to intermediate work. Review arithmetic and geometric sequences first. OpenStax's sequence lesson gives more practice writing a rule from terms.

A basic idea: make a second row

For $2,5,10,17,26$, subtract each term from the next. The first differences are $3,5,7,9$. They are not constant, so an arithmetic rule does not fit. Subtract again: the second differences are $2,2,2$. A quadratic is a good candidate. Here $a_n=n^2+1$ gives all five listed terms when $n$ starts at $1$.

A quick tour of pattern clues

These tests assume equally spaced positions: term 1, term 2, term 3, and so on. For $a_n=An^2+Bn+C$, the second difference is $2A$ when the index rises by $1$. So a second difference of $2$ suggests $A=1$; the first two terms then determine $B$ and $C$.

For a ratio, divide each term by its predecessor. The sequence $4,6,9,13.5$ has ratio $1.5$ each time, although its differences are $2,3,4.5$. A changing difference can still come from a very simple multiplier.

A finite list never proves a unique continuation. Many formulas can match its first five terms and differ later. Use the story behind the data, a stated rule, or extra terms to choose a model. Differences and ratios are clues, not a substitute for checking the proposed rule.

Application 1: square tiles with one extra

A pattern uses $2,5,10,17,26$ tiles in designs $1$ through $5$. Each design is an $n$ by $n$ square plus one extra tile. Predict design $6$ and explain why.

1. Read the structure. Design $n$ has $n^2$ square tiles and $1$ extra, so $a_n=n^2+1$.
2. Check the list. $a_1=2$, $a_2=5$, and $a_5=26$ agree.
3. Predict. $a_6=6^2+1=37$. The differences predict the same result: the next first difference is $11$, and $26+11=37$.

The picture of the design supplies a reason for the formula; the table alone would not.

Application 2: a repeated multiplier

A lab model records $16,24,36,54$ units at four equally spaced observations. If the same multiplicative rule continues, what is the next model value?

1. Compare ratios. $24/16=36/24=54/36=3/2$. The differences, $8,12,18$, are not constant.
2. Write the candidate rule. With observation 1 as the start, $a_n=16(3/2)^{n-1}$.
3. Predict and check. $a_5=54(3/2)=81$. Substituting $n=4$ gives $16(3/2)^3=54$, matching the last known value.

This prediction is conditional on the model continuing; observations could change for reasons the short table does not show.