Sequences

The numerical pattern below is given: Row 1 $\frac{1}{1\times2}=\frac12$, Row 2 $\frac{1}{2\times3}=\frac12-\frac13$, Row 3 $\frac{1}{3\times4}=\frac13-\frac14$, Row 4 $\frac{1}{4\times5}=\frac14-\frac15$.

May/June 2015

The first four terms in the sequence $S$ are $89, 83, 77, 71$.

May/June 2015

A chain of diagrams is built with counters.

May/June 2016

The $n$th term of a sequence is $n^{2}-5n$. Determine the $2$nd term of the sequence.

May/June 2016

The diagrams illustrate designs built from crosses (X) and circles (O).

May/June 2017

The first four terms in the sequence are $u_1,u_2,u_3,u_4$.

May/June 2018

Find an expression, in terms of $n$, for the $n$th term in this sequence $1,\,7,\,13,\,19,\,25,\ldots$

May/June 2018

The expression for the $r$th term of the sequence, $u_r$, is $u_r = 3r + 2$.

May/June 2019

The initial three dot-and-line patterns in the sequence are shown here.

May/June 2019

For this sequence, the term-to-term rule is: multiply by $2$ and add $3$. The first three terms are $1, 5$ and $13$. Write down the next term in this sequence.

May/June 2021

The sequence begins with the four terms $\frac{12}{16}, \frac{17}{25}, \frac{22}{36}, \frac{27}{49}$.

May/June 2021

Shani creates a run of patterns with counters.

May/June 2022

Shown below are the first four patterns in a sequence formed from grey tiles and white tiles.

May/June 2022

The sequence starts with these four terms: $1, 7, 13, 19$.

May/June 2023

The initial four terms of a sequence are $1, 3, 9, 27$. Find the next term in the sequence.

May/June 2023

The diagrams illustrate the first four arrangements in the sequence.

May/June 2024

The first four patterns belong to a sequence built from black beads and white beads.

May/June 2025

The first four terms of a sequence are $16, 13, 10, 7$.

May/June 2025

The diagrams present a run of triangles constructed from identical sticks. Compared with the triangle before it, each one has two extra sticks on every edge.

Oct/Nov 2016

The first three terms of two sequences are $1,3,5$.

Oct/Nov 2016

Each term in the sequence is $4$ greater than the one immediately before it. The third term is $17$, and the fourth term is $21$.

Oct/Nov 2018

The first five terms of a sequence are given as $\frac{3}{4}, \frac{7}{8}, \frac{11}{12}, \frac{15}{16}, \frac{19}{20}$.

Oct/Nov 2018

The first four patterns in a counters sequence are displayed.

Oct/Nov 2019

The diagram displays a sequence of patterns. Each pattern contains one additional row, and two extra dots in every row, compared with the previous pattern.

Oct/Nov 2020

The sequence begins with the five terms $4, 8, 16, 32, 64$.

Oct/Nov 2021

The first three patterns in a counter-based sequence are shown here.

Oct/Nov 2021

A sequence has $n$th term $3n^2 - 1$. Find the first three terms of the sequence.

Oct/Nov 2022

A linear sequence has second term $28$ and fifth term $16$. Find the first term, the third term and the fourth term of the sequence.

Oct/Nov 2023

These are the first five terms in a sequence: $5,\ 12,\ 19,\ 26,\ 33$. Find the next term in the sequence.

Oct/Nov 2023

The sequence begins with these four terms: $2, 8, 14, 20$.

Oct/Nov 2024

A series of patterns is formed using crosses and circles.

Oct/Nov 2024

These are the first four dot patterns in the sequence.

Oct/Nov 2025

The first five terms of a sequence are: 3, 8, 13, 18, 23. Determine the next term in the sequence.

Oct/Nov 2025