Math Core

"What is the $100$ th term?" "What is the $2026$ th letter?" "How many dots are in the next figure?" Contest writers love patterns because a little insight replaces a lot of counting. This module covers the patterns you'll meet most: arithmetic sequences, repeating cycles, and growing figures.

Arithmetic sequences

An arithmetic sequence adds the same number, the common difference $d$ , each time: $7, 11, 15, 19, \dots$ has $d = 4$ .

To get from the $1$ st term to the $n$ th term, you take $n - 1$ steps of size $d$ .

Arithmetic sequence formulas

$n$ th term: $a_n = a_1 + (n - 1)d$
Number of terms from $a_1$ to $a_n$ : $n = \dfrac{a_n - a_1}{d} + 1$
Sum of the terms: $S = n \cdot \dfrac{a_1 + a_n}{2}$ (the number of terms times the average of the first and last)

The sum formula works because pairing the first and last terms, the second and second-to-last, and so on gives equal pairs, the same trick as $1 + 2 + \dots + 100$ .

Common mistake

The fence-post error: from $5$ to $20$ counting by ones there are $16$ numbers, not $15$ . Dividing the distance by the step counts the gaps; add $1$ to count the terms.

If you know two terms that aren't next to each other, the number of steps between them tells you $d$ . If $a_3 = 17$ and $a_{10} = 52$ , there are $7$ steps covering $35$ , so $d = 5$ .

Geometric sequences

A geometric sequence multiplies by the same number $r$ each time: $3, 6, 12, 24, \dots$ has $r = 2$ , and $a_n = a_1 \cdot r^{n-1}$ . These grow fast, so questions like "which term first passes $1000$ ?" can be done by just listing terms.

Repeating patterns

If a pattern repeats every $p$ terms, then the $n$ th term depends only on the remainder when $n$ is divided by $p$ . A remainder of $0$ means the last item in the block. Days of the week repeat every $7$ , decimal expansions of fractions repeat, and many recursively defined sequences turn out to cycle. When a sequence is defined by a rule, write out terms until they repeat.

Growing figures

For picture patterns, count the first few figures and look at the differences between counts.

Constant differences mean an arithmetic sequence (a linear formula).
Differences that themselves go up by a constant (second differences constant) mean a formula with $n^2$ .

Worked example: Counting terms and summing

How many terms are in $7, 11, 15, \dots, 203$ , and what is their sum?

Number of terms: $\dfrac{203 - 7}{4} + 1 = 49 + 1 = 50$ . Sum: $50 \cdot \dfrac{7 + 203}{2} = 50 \cdot 105 = 5250$ .

Worked example: A repeating word

The letters of COUNT are written over and over: COUNTCOUNTCOUNT… What is the $2026$ th letter?

The block has $5$ letters. Since $2026 = 5 \cdot 405 + 1$ , the $2026$ th letter is the $1$ st letter of the block: C.

Worked example: Toothpick squares

A row of $1$ square uses $4$ toothpicks, a row of $2$ squares uses $7$ , and a row of $3$ uses $10$ . How many toothpicks make a row of $50$ squares?

Each new square adds $3$ toothpicks (it shares one side with the square before). So the count is $4 + 3(n - 1) = 3n + 1$ , and $50$ squares need $151$ toothpicks.

Worked example: Second differences

What is the $20$ th term of $2, 5, 10, 17, 26, \dots$ ?

The differences are $3, 5, 7, 9$ , which go up by $2$ , so the formula involves $n^2$ . Compare with the squares $1, 4, 9, 16, 25$ : each term is one more. So $a_n = n^2 + 1$ and $a_{20} = 401$ .

Tip

To find a formula for a pattern, test it on at least three known terms. Two terms fit lots of wrong formulas.