Sequences

The order matters. The 1st term is 3, the 2nd term is 7, and so on. The number that says where a term sits is its position.

The pattern is the heart of a sequence. Once you spot the pattern, you can continue the list, fill in gaps, and even leap straight to a term far down the line. The rest of this guide is about describing that pattern in two clever ways.

• A sequence is an ordered list of numbers.
• Each number in the list is a term.
• The order matters — the 1st term, 2nd term, and so on.
• Spotting the pattern lets you continue the sequence.

To continue a sequence with a term-to-term rule, you just keep applying it. To take $3, 7, 11, 15$ further, add 4: the next terms are 19, 23, 27.

A term-to-term rule always needs a starting term. "Add 4" on its own does not pin down a sequence — you must also know it begins at 3.

There is one catch. A term-to-term rule is great for the next term, but slow for a far-off one. To find the 50th term of $3, 7, 11, \ldots$ you would have to add 4 forty-nine times. That is exactly the problem the next idea solves.

• A term-to-term rule steps from one term to the next.
• It can use any operation: add, subtract, multiply or divide.
• You also need the starting term.
• It is slow for terms far down the sequence.

Take the rule $\text{term} = 3n + 2$. To find a term, substitute its position for $n$:

• 1st term: $3 \times 1 + 2 = 5$.
• 2nd term: $3 \times 2 + 2 = 8$.
• 3rd term: $3 \times 3 + 2 = 11$.

This is the power of a position-to-term rule. Want the 50th term? Just put $n = 50$: $3 \times 50 + 2 = 152$. One quick calculation, no listing.

A position-to-term rule is really a function — the position goes in, the term comes out.

• A position-to-term rule finds a term from its position $n$.
• Substitute the position number in place of $n$.
• It jumps straight to any term, however far away.
• It is faster than a term-to-term rule for distant terms.

1. Find the common difference: it goes up by 3 each time, so the rule starts with $3n$.
2. List what $3n$ gives: $3, 6, 9, 12, \ldots$
3. Compare with the real sequence: each term is 2 more than $3n$.
4. So the nth-term rule is $3n + 2$.

The shortcut: the common difference is the number in front of $n$, then you add or subtract a number to match the first term.

Check it by putting $n = 1$: $3 \times 1 + 2 = 5$ — the correct first term.

• A linear sequence has a constant common difference.
• The common difference is the number in front of $n$.
• Compare $dn$ with the sequence, then add or subtract to match.
• Always check the rule on the first term.

Square, triangular and cube numbers are not linear — their differences are not constant, so they do not have a simple $dn + c$ rule. Recognising them quickly will save you a lot of time.

• Square numbers: $1, 4, 9, 16, \ldots$ — rule $n^2$.
• Triangular numbers: $1, 3, 6, 10, \ldots$ — growing gaps.
• Cube numbers: $1, 8, 27, 64, \ldots$ — rule $n^3$.
• Fibonacci: each term is the sum of the two before it.

Step 3

Check whether the terms go up by adding. $6 - 3 = 3$ but $12 - 6 = 6$ — the gap is not constant.

1. Step 1

   Look at the differences between terms: $3, 5, 7, 9, \ldots$ — they increase by 2 each time.

2. Step 2

   The next difference is $9 + 2 = 11$, so add 11 to the last term.

   $28 + 11 = 39$

3. Step 3

   The sequence is not linear because the differences are not constant — they keep growing.

4. Step 4

   Since there is no single common difference, a simple $dn + c$ rule cannot describe it.