Number sequences and patternsIB MYP Maths Extended: Revision notes

A sequence is a list of numbers that follow a rule. Each number is a term. You can describe a rule in two ways:

• Term-to-term rule: how to get from one term to the next, such as "add 4".
• Position-to-term rule (the $n$th term): how to get a term from its position $n$, such as $4n+3$. To continue a sequence, look at the differences between neighbouring terms. For $7,11,15,19$ the differences are all $4$.

Some sequences have names you should recognise.

• Square numbers: $1,4,9,16,25,\ldots$ ($n^2$).
• Triangular numbers: $1,3,6,10,15,21,\ldots$. The differences are $2,3,4,5,6,\ldots$, so each term adds the next whole number: $1$, $1+2$, $1+2+3$, and so on.
• Cube numbers: $1,8,27,64,\ldots$ ($n^3$).
• Fibonacci-type: each term is the sum of the previous two. $1,1,2,3,5,8,13,\ldots$ If the first two terms are different, such as $2,3,5,8,13$, it is still Fibonacci-type.

A sequence with a common difference $d$ has $n$th term $an+b$, where $a$ equals the common difference and $b$ is found from the first term: $b=\text{first term}-a$. Example: $7,11,15,19$ has $d=4$, so the rule starts $4n$. $4(1)=4$ but the first term is $7$, so add $3$: the $n$th term is $4n+3$. For a decreasing sequence $d$ is negative: $20,17,14$ has $d=-3$ and $n$th term $-3n+23$, which can be written $23-3n$.

Finding a term: substitute the position. For $4n+3$, the 50th term is $4(50)+3=203$. Testing if a number is in the sequence: set the rule equal to the number and solve for $n$. The number is a term only if $n$ is a positive whole number.

• Is $150$ in $4n+3$? $4n+3=150$ gives $n=36.75$. Not whole, so no.
• Is $-40$ in $23-3n$? $23-3n=-40$ gives $n=21$. Whole, so yes: it is the 21st term. First term above or below a value: solve the inequality, then round to the next whole number. $23-3n<0$ gives $n>7.67$, so $n=8$.

Pattern questions often describe a diagram, such as tables pushed together in a row. To generalise the pattern:

1. Count the value for the first few patterns and put them in a table: $4,6,8$ seats for $1,2,3$ tables.
2. Find the common difference ($2$), so the rule starts $2n$.
3. Use the first term to find $b$: $2(1)+b=4$ gives $b=2$, so the rule is $2n+2$.
4. Explain the rule using the diagram: each table adds a seat on the top and bottom ($2n$), and each end of the row adds one more ($+2$).
5. Verify with a new case, counting directly: six tables give $6+6+2=14$, and the rule gives $2(6)+2=14$.

Real situations can use the same rule. For the tables, $2n+2$ seats for $n$ tables. To seat $31$ people solve $2n+2\ge31$: $n\ge14.5$, so use the next whole number, $15$ tables. Always round to a whole number that still meets the need, and state any assumption of the model, such as one person sitting at each free edge.