Sequences

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 sequence is an ordered list of numbers called terms.
• Each term has a position; the term in position $n$ is $T_n$.
• A term-to-term rule jumps from one term to the next.
• A position-to-term rule gives any term directly from $n$.

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.

• Linear sequences have a constant common difference $d$.
• The nth term is $T_n = dn + (a - d)$.
• Find $d$, then the zeroth term, then write the rule.
• Always check by plugging in $n = 1$.

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 $T_n = n^2 + 2n$.

• Constant second differences mean a quadratic sequence.
• If the constant second difference is $\Delta_2$, then $a = \Delta_2 / 2$.
• Subtract $an^2$ from each term to leave a linear sequence.
• Handle the leftover linear sequence using your usual method.

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.

• Count, write the sequence, then find differences.
• Use the nth-term rule rather than drawing far-off patterns.
• Picture questions often hide linear or simple quadratic sequences.
• Triangular numbers $1, 3, 6, 10, \ldots$ are a classic quadratic example.

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.

• Substitute the position into the rule to get any term directly.
• Set the rule equal to a value and solve to find its position.
• A non-integer answer for $n$ means the value isn't in the sequence.
• Use inequalities for 'first term less than…' style questions.

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.

• Linear graphs use the same algebra as linear sequences.
• Quadratic sequences mirror quadratic functions and graphs.
• Real-life saving and growth plans are sequence problems.
• Confidence with $T_n$ unlocks lots of later algebra.

The common difference is $d = 3$.

Step 1