Sequences

A sequence is just an ordered list of numbers. Each number in the list is called a term, and each term has a position — first, second, third, and so on. The first term is usually labelled $T_1$ or $a$, the second $T_2$, the third $T_3$, and the term in position $n$ is written $T_n$.

The rule that ties a sequence together can be described two ways. A term-to-term rule says how to get from one term to the next, like "add $3$" or "double then subtract $1$". A position-to-term rule (also called the nth-term rule) gives any term directly from its position, like $T_n = 3n + 2$.

For the sequence $5, 8, 11, 14, 17, \ldots$ the term-to-term rule is "add $3$", and the position-to-term rule is $T_n = 3n + 2$. Both describe the same pattern, but the nth-term rule is more useful when you need a far-off term like $T_{100}$ — no need to add $3$ ninety-nine times!

A linear sequence (sometimes called an arithmetic sequence) has a constant gap between consecutive terms. That gap is called the common difference, written $d$.

The nth-term rule for a linear sequence is

$T_n = d \, n + (a - d)$

where $a$ is the first term and $d$ is the common difference.

Here's the friendly recipe.

1. Find $d$ — the constant difference between terms.
2. Find the zeroth term $(a - d)$ — work backwards one step from $T_1$.
3. Write the rule as $T_n = dn + (a - d)$.
4. Check with $n = 1$.

For $5, 8, 11, 14, \ldots$: $d = 3$, the zeroth term is $5 - 3 = 2$, so $T_n = 3n + 2$. Check: $T_1 = 3(1) + 2 = 5$. ✓

For a decreasing sequence like $20, 17, 14, 11, \ldots$ the common difference is $-3$, the zeroth term is $20 - (-3) = 23$, and $T_n = -3n + 23$. So $T_{10} = -3(10) + 23 = -7$. A quick double-check beats a clever trick every time.

A quadratic sequence has an $n^2$ term hidden inside it: $T_n = an^2 + bn + c$. The clue is the second differences. The first differences grow (or shrink) at a constant rate, and the difference of the differences — the second difference — is constant.

Here is the link to remember: if the constant second difference is $\Delta_2$, then $a = \dfrac{\Delta_2}{2}$.

For $1, 4, 9, 16, 25, \ldots$ the first differences are $3, 5, 7, 9$ and the second differences are all $2$. So $a = 1$, and a quick check gives the famous $T_n = n^2$.

A friendly recipe for simple quadratics:

1. Write out the first differences and the second differences.
2. If the second differences are constant, set $a = \dfrac{\Delta_2}{2}$.
3. Write out the values of $an^2$ for $n = 1, 2, 3, \ldots$.
4. Compare with the original terms — the leftover is a linear sequence, which you handle as before.

For $3, 8, 15, 24, 35, \ldots$ second differences are $2$, so $a = 1$. Subtract $n^2$ from each term: $3-1=2$, $8-4=4$, $15-9=6$, $24-16=8$ — the leftover sequence is $2n$. So $T_n = n^2 + 2n$.

Many sequence questions don't give you a list of numbers at all — they show a pattern of dots, matchsticks or coloured tiles, and ask you to find a rule.

The friendly recipe is the same.

1. Count carefully how many objects make patterns $1$, $2$, $3$, $4$.
2. Write the sequence of counts.
3. Find the differences and use them to write the nth-term rule.
4. Use the rule to answer the question (often "how many in pattern $20$?").

For a row of squares with $4, 7, 10, 13, \ldots$ matchsticks, the common difference is $3$, the zeroth term is $4 - 3 = 1$, and $T_n = 3n + 1$. So pattern $20$ needs $3(20) + 1 = 61$ matchsticks. Now the rule does all the work — you never have to draw pattern $20$.

For a triangular array of dots like $1, 3, 6, 10, 15, \ldots$ first differences are $2, 3, 4, 5$ — not constant — but second differences are all $1$. So it is a quadratic sequence with $a = \tfrac{1}{2}$, and indeed $T_n = \tfrac{1}{2}n(n+1)$ — the triangular numbers.

Once you've written down a position-to-term rule like $T_n = 3n + 2$, two kinds of question become easy.

Find a specific term. Substitute the position number for $n$. For $T_n = 3n + 2$, the 25th term is $T_{25} = 3(25) + 2 = 77$. No counting needed.

Find which position gives a particular value. Treat the rule as an equation and solve. "Is $50$ in the sequence $5, 8, 11, 14, \ldots$?" Set $3n + 2 = 50$, giving $3n = 48$ and $n = 16$. Since $n = 16$ is a whole number, yes — $50$ is the 16th term. If $n$ came out as a fraction, the number is not in the sequence.

For a decreasing sequence like $T_n = 100 - 4n$, "What is the first term less than $0$?" Solve $100 - 4n < 0$, giving $n > 25$. So $T_{26} = 100 - 4(26) = -4$ is the first term below zero. The nth-term rule turns every sequence question into a tidy bit of algebra.

Sequences may look like a stand-alone topic, but they tie into a lot of later maths:

• Linear graphs use the same $T_n = mn + c$ idea — gradient is the common difference.
• Quadratic graphs mirror quadratic sequences: $y = ax^2 + bx + c$ uses the same coefficients.
• Spreadsheets and coding lean on sequences whenever a loop runs over $n = 1, 2, 3, \ldots$.
• In real life, sequences model saving plans, repeated patterns and growing structures.

If a future topic feels new, ask whether you're really just looking at a sequence in disguise. Master differences and the nth-term rule now and you'll thank yourself later.