The nth term

Sara and Tom both make a sequence of numbers. The first term in both sequences is 10.

The term-to-term rule for Sara’s sequence is add 14. The term-to-term rule for Tom’s sequence is add 35.

Each sequence stops when a term reaches or goes above 500.

Find the number of terms that are the same in both Sara’s and Tom’s sequence. You must show your working.

Find an expression for the n th term of Sara’s sequence.

A sequence is defined by

Find the value of k.

Another sequence is defined by u_n+1 = 6u_n - 7.

Find the value of u₁ in the case when u₂ = 2 × u₁.

Explain what happens to the sequence u_n+1 = 6u_n - 7 when u₁ = 1.4.

Sara and Tom both make a sequence of numbers. The first term in both sequences is 10.

The term-to-term rule for Sara’s sequence is add 14. The term-to-term rule for Tom’s sequence is add 35.

Each sequence stops when a term reaches or goes above 500.

Find the number of terms that are the same in both Sara’s and Tom’s sequence. You must show your working.

Find an expression for the n th term of Sara’s sequence.

A sequence is generated using the rule:

• multiply the previous term by 3

• then subtract 1.

The 2nd term of the sequence is 20.

Find the 3rd term of the sequence.

Find the 1st term of the sequence.

Here are the first four terms of a different sequence.

5 10 15 20

Find the n th term of the sequence.

Here are the first three tile patterns of a sequence.

Pattern 1 Pattern 2
Pattern 3
Pattern 4

Draw Pattern 4 in the space above.

Complete this table.

Each pattern in the sequence can be split into a square of tiles and a single column of tiles.

For example, Pattern 3:

The square in Pattern n contains 4096 tiles.

Work out how many tiles are in Pattern n.

Some sequences are defined using this term-to-term rule.

u_n+1 = 5u_n - 8.

If u₃ = 22, show that u₄ = 102.

If u₃ = 22, work out u₂.

If u₁ = 2, write down the value of u₅₀. Give a reason for your answer.

A sequence has nth term 2n² + 1.

Prove algebraically that the sum of any two consecutive terms in this sequence is always a

multiple of 4.

Here are the first three dot patterns in a sequence.

Pattern 1
Pattern 2
Pattern 3

Draw Pattern 4 in the sequence.

Without drawing, work out how many dots are in Pattern 10 of the sequence. Explain how you worked out your answer.

Here are the first five terms of a sequence.

4 8 12 16 20

Write down the next term of the sequence.

Kai says

All of the terms in the sequence are even numbers. 402 is an even number. Therefore, if the sequence is continued, 402 will be in the sequence.

Is Kai correct? Give a reason for your decision.

Here are the first five terms of another sequence.

0.25 0.5 1 2 4

Write down the next term of the sequence.

Explain how you worked out your answer.

A sequence is defined by

Work out the value of u₂ and the value of u₃.

Here are the first four terms of a quadratic sequence.

- 2 7 22 43

The sequence has the formula x_n = an² + b.

Find the value of a and the value of b.

Find the next term of this sequence.

1 3 9 27 81........

In the Fibonacci sequence below, the next term is found by adding the two previous terms. The third term is 0.83 and the fourth term is 1.29.

Complete the first, second and fifth terms of the sequence.

Here are the first four terms of a sequence.

5 12 19 26

Write down the next term in the sequence.

Explain how you worked out your answer.

Ryan started with 80 counters.

Ryan says

I still have enough counters to make Pattern 5 and Pattern 6.

Is Ryan correct? Show how you decide.

Complete the table below for the addition of counters in consecutive patterns.

The number of counters in Pattern k + Pattern (k + 1) is 144.

Find the value of k.

Here are the first four terms of a sequence.

2 5 8 11 5 10 17 26

Find the next term.

Find the n th term.

Here are the first four terms of a sequence.

7 15 23 31

Write down the next term of the sequence.

Explain how you worked out your answer.

Explain why 80 is not a term in this sequence.

The next term in a Fibonacci sequence is found by adding together the two previous terms.

The first and second terms of a particular Fibonacci sequence are x and y.

Show that the fourth term of the sequence can be written as x + 2 y.

The fourth term of the same Fibonacci sequence is 7. The seventh term of the sequence is 31.

Work out the value of x and the value of y. You must show your working.

Here are the first four terms of a sequence.

Write an expression for the n th term.

Here are the first four terms of a quadratic sequence.

– 1 5 13 23

The nth term is n² + bn + c.

Find the value of b and the value of c.

Here are the first four dot patterns in a sequence.

Pattern 1
Pattern 2
Pattern 3
Pattern 4

Draw Pattern 5 in the sequence.

Without drawing, work out how many dots are in Pattern 8 of the sequence. Explain how you worked out your answer.

Here are the first four terms of a sequence.

8 15 22 29

Write down the next term in the sequence.

Explain how you worked out your answer.

The n th term of a different sequence is given by 4 n + 2.

Explain why 32 is not a term in this sequence.

Here are the first four terms of a quadratic sequence.

- 1 3 13 29

The nth term is an² + bn + c.

Find the values of a, b and c.

Complete the first seven square numbers.

1 4 9 16........ 36 49

Write the missing term in each sequence.

18 16 14........ 10 8

........ 14 20 26 32 38

Here is a sequence.

Work out the next term.

Find the n th term.

A sequence is defined by the rule u_n+1 = 5u_n - 15.

If u₃ = 6, calculate

u₅

u₂

Trevor says

Show that Trevor is correct.

Here are the first four terms of a sequence.

3 8 13 18

Write down the next term of the sequence.

Explain how you worked out your answer.

Explain why 534 is not a term in this sequence.

Here are the first three patterns in a sequence.

Pattern 1
Pattern 2
Pattern 3

Draw Pattern 4 in the sequence.

Without drawing it, work out how many dots there are in Pattern 8. Explain how you decide.

Pattern n has 196 dots.

Find the value of n.

Here are the first four terms of a sequence.

- 1 4 9 14

Write an expression for the n th term of this sequence.

The n th term of another sequence is given by

The third term is 9 and the sixth term is 126.

Find the value of a and the value of b.

Here are the first four terms of a sequence.

28 23 18 13

Find the n th term of the sequence.

Here are the first four terms of a quadratic sequence.

2 15 34 59

The nth term is an² + bn + c.

Find the values of a, b and c.

Here is a sequence.

Work out the next term.

Find the n th term.

Here are the first four terms of a sequence.

2 4 8 16

What is the next term in the sequence?

Explain how you worked out your answer.

Here are the first four terms of a sequence.

6 10 14 18

Write down the next term.

Write an expression for the n th term.

Explain why 511 is not a term in the sequence.

Find the term in the sequence that is nearest to 511.

Here are the first four terms of a quadratic sequence.

0 9 22 39

The nth term can be written as an² + bn + c.

Find the values of a, b and c.

Write the next term in each of these sequences.

1 1 2 3 5 8

2 4 8 16 32 64

Write an expression for the n th term of the sequence below.

15 12 9 6

Write the next term in each of these sequences.

1 1 2 3 5 8

2 4 8 16 32 64

Write an expression for the n th term of the sequence below.

15 12 9 6

A sequence is defined using this term-to-term rule.

Another sequence is defined using this term-to-term rule,

where k and r are constants.

Given that u₂ = 41, u₃ = 206 and u₄ = 1031, find the value of k and the value of r.

Write down the second and third terms of the following two sequences.

Rule: To get the next term subtract 4 from the previous term. First term = 19.

Rule: To get the next term multiply the previous term by 5 and then add 3. First term = 7.

Here are the first four terms of another sequence.

5 9 13 17

Write an expression for the n th term of this sequence.

Here are the first four terms of a sequence.

Find the nth term of this sequence.

Here are the first four terms of a quadratic sequence, the n th term of this quadratic sequence is an² + bn + c.

2 12 28 50

Find the values of a, b and c.