Patterns in Mathematics: Solving Pattern-Based Questions: 17 MCQs with answers

A · Find the rule that generates every term

A pattern is a relationship between terms; solving it means finding the rule that generates every term, not just guessing the next one.

A · $4, 4, 4$

Subtract consecutive terms: $9-5=4$, $13-9=4$, $17-13=4$. The constant first difference is $4$.

D · Arithmetic (linear)

Constant first differences are the signature of an arithmetic (linear) pattern, where each term is obtained by adding a fixed number.

A · Divide each term by the previous one

Geometric patterns have a constant ratio between consecutive terms, so you divide each term by the one before it.

B · $3, 6, 12, 24, \dots$

In $3, 6, 12, 24, \dots$ each term is double the previous one, so the ratio is constant at $2$.

B · A geometric pattern

A constant ratio means each term is a fixed multiple of the previous one, which defines a geometric pattern.

A · True

In an arithmetic pattern, the $n$-th term is given by $T_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

A · All equal to $2$

First differences are $3, 5, 7, 9$; subtracting again gives $2, 2, 2$. Constant second differences confirm a quadratic pattern.

C · The third term shows the rule is 'multiply by $2$'

Testing against every term: $2+2=4$ works, but $4+2=6 \neq 8$. The rule 'multiply by $2$' works for all terms.

B · $a + (n-1)d$

The $n$-th term of an arithmetic sequence is $T_n = a + (n-1)d$: start at $a$ and add $d$ exactly $n-1$ times.

D · A rule that fails a later term is not the correct rule

Many different rules fit the first two terms. Only a rule that works for every given term is the correct one.

B · Check that adding $4$ gives every term from the previous one

A rule is verified by testing it against every given term: $5+4=9$, $9+4=13$, $13+4=17$. All hold, so the rule is correct.

C · Quadratic

First differences are not constant, but second differences are constant ($2$), which is the signature of a quadratic pattern.

D · $42$

First differences are $4, 6, 8, 10$; the next difference is $12$, so the next term is $30+12=42$. The rule is $n(n+1)$.

C · It can reveal the pattern type even for unfamiliar sequences

The difference-table method is systematic: by computing first and second differences you can classify the pattern even when you do not recognise the sequence.

B · Writing a rule compactly is the same skill as writing an algebraic expression

Describing a sequence with a rule is exactly the skill of writing an algebraic expression for an unknown quantity, which is the core of algebra.

C · 50

To find the rule, first calculate the first differences: $5, 7, 9, 11$. The first differences are not constant, so check the second differences: $2, 2, 2$. Since the second differences are constant, the pattern is quadratic. The rule can be expressed as $n(n+2)$, where $n$ is the term index starting from 1. Thus, the n…