IEB Grade 11 Number Patterns Cheat Sheet

Patterns are about 25 marks of Grade 11 Paper 1. The whole topic fits on one page.

Recognise the pattern

• Constant first difference: linear, $T_n=dn+c$.

• Constant second difference: quadratic, $T_n=a{n}^2+bn+c$.

Finding $T_n$ for a quadratic pattern

• $2a$ = second difference

• $3a+b=T_2-T_1$

• $a+b+c=T_1$

• Check the formula with $T_3$.

Using $T_n$

• A term: substitute $n$.

• A term number: solve $T_n=$ value;
  $n$ must be a natural number.

• Is a value in the pattern? Only if $n$ comes out as a natural number.

• First differences: $T_{n+1}-T_n$ is linear.

Patterns in context

• Count the first three or four diagrams carefully, then find differences.

• Maximum or minimum term: $n=-\frac{b}{2a}$, rounded to the nearest natural number if needed.

• A quadratic pattern with $a>0$ eventually overtakes any linear pattern.

Know by heart

• The three equations for $a$, $b$ and $c$

On the formula sheet

• Nothing for quadratic patterns

Common traps

• Taking $a$ as the second difference instead of half of it.

• Keeping a negative or fractional $n$.

• Miscounting the first diagram in a context question.