GCSE Maths Sequences & nth term questions by topic

Generating terms.

1. (a)A sequence starts at 3 with rule “add 4”. Write the first four terms.[1]
2. (b)Write the first four terms of the sequence with nth term 6n − 2.[2]
3. (c)A sequence starts at 48 with rule “halve”. Write the next two terms.[1]

Finding the nth term.

1. (a)Find the nth term of 7, 10, 13, 16, …[2]
2. (b)Use it to find the 15th term.[1]
3. (c)Is 40 a term of this sequence? Give a reason.[1]

Special sequences.

1. (a)Write the next two terms of 1, 4, 9, 16, … and name the sequence.[2]
2. (b)Write the next two terms of 1, 3, 6, 10, 15, …[2]

From rules to lists.

1. (a)A sequence starts at 100 with rule “subtract 7”. Write the first three terms.[1]
2. (b)Will any term of that sequence equal 80? Explain.[2]
3. (c)Write the first three terms with nth term n² + 1.[1]

A theatre seats rows of increasing length.

1. (a)Row 1 has 14 seats and every row after it has 2 more. Work out the number of seats in row 6.[1]
2. (b)Show that the number of seats in row n is 2n + 12.[2]
3. (c)A row must be built with exactly 40 seats. Work out which row that is.[2]

Mixed generation and squares.

1. (a)Write down the first three terms of the sequence with nth term 100 − 7n.[2]
2. (b)From the list 49, 60, 81 and 95, write down the square numbers.[2]

Membership and positions.

1. (a)Find the nth term of 9, 13, 17, 21, …[2]
2. (b)Is 101 a term of the sequence?[2]
3. (c)Which term of the sequence equals 45?[1]

Patterns in context.

1. (a)A theatre’s rows have 12, 15, 18, 21 seats. Find the nth term.[2]
2. (b)How many seats are in the 20th row?[1]
3. (c)Does a row with exactly 50 seats exist?[1]

Fibonacci-type work. (Higher)

1. (a)Write the next three terms of 2, 5, 7, 12, 19, …[2]
2. (b)A Fibonacci-type sequence starts a, b. Write expressions for the 3rd, 4th and 5th terms.[3]

Geometric progressions. (Higher)

1. (a)Write the next two terms of 5, 15, 45, … and state the ratio.[2]
2. (b)Write the next two terms of 2, 2√3, 6, …[2]

A membership test with reasoning.

1. (a)A sequence has nth term 7n − 4. Write down its first three terms.[1]
2. (b)Decide, with working, whether the value 150 belongs to the sequence.[2]
3. (c)Decide, with working, whether the value 100 belongs to the sequence.[2]

Quadratic nth term. (Higher)

1. (a)Find the nth term of 5, 8, 13, 20, 29, …[3]
2. (b)Find the nth term of 3, 9, 19, 33, 51, …[3]

Is it a term? (Higher)

1. (a)A quadratic sequence has nth term n² + 2n. Is 48 a term?[2]
2. (b)Is 80 a term?[2]
3. (c)For the linear sequence 6n − 4, is −28 a term?[1]

Mixed sequences. (Higher)

1. (a)A sequence has nth term n² − 3n. Write the first five terms.[2]
2. (b)Describe the pattern of the terms’ signs.[1]
3. (c)Find the nth term of the sequence of first differences of n² − 3n.[3]

A geometric investigation. (Higher)

1. (a)A geometric progression starts 729, 243, 81. Find the ratio and the next three terms.[3]
2. (b)Which term of the progression equals 3? Show your reasoning.[2]
3. (c)Explain why no term of this progression equals 40.[2]

Fibonacci and geometric together. (Higher)

1. (a)A Fibonacci-type sequence runs 3, 4, 7, 11, 18, … Write down the next term.[1]
2. (b)A geometric sequence starts 2, 6, 18, … Write down its nth term.[3]
3. (c)Work out the 10th term of your geometric sequence.[1]

A quadratic sequence is decoded. (Higher)

1. (a)A sequence runs 4, 10, 18, 28, 40, … Show that its second differences are constant and state their value.[2]
2. (b)Show that the sequence has nth term n² + 3n, and check your formula on the third term.[3]
3. (c)Work out the 20th term.[1]

A doubling investigation. (Higher)

1. (a)A bacteria culture starts at 500 cells and doubles every 20 minutes. Write an expression for the population after n twenty-minute intervals.[2]
2. (b)Work out the population after 2 hours.[2]
3. (c)Explain why the population can never be exactly 10 000 cells under this model.[2]