Special sequences

Write down an odd number between 10 and 20.

Write down two cube numbers less than 10.

Write down the 4th term of this sequence of numbers.

1 4 9........ 25 36 49

Write down the name for the sequence of numbers.

Amit draws these four diagrams. Amit writes the number of dots on the perimeter underneath each diagram.

Number of dots on the perimeter 1 3 6 9

Amit says

My diagrams show the first four triangular numbers.

Complete this statement.

Amit is incorrect because........

Use the grid below to draw the correct diagram for the 5th triangular number.

Rosa is thinking of a fraction. The numerator is a cube number less than 100. The denominator is a square number less than 100.

Find the fraction that Rosa is thinking of.

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.

Work out the next term in this sequence.

1 3 6 10........

In the Fibonacci sequence below, the next term is found by adding the two previous terms. The second term is 7, the third term is 10 and the fourth term is 17.

Work out the first and fifth terms of the sequence.

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.

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.

The kth term of a sequence is r^k, where r ≠ 0. The sixth term is equal to three times the second term.

Find the value of r, giving your answer correct to 3 decimal places.

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.

Write down a factor of 18.

Write down a square number between 10 and 20.

Find the two numbers which multiply together to make 40 and add together to make 13.

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 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.

The next term in each of these Fibonacci sequences is found by adding together the two previous terms. Work out the missing terms in each sequence.

2 5 7 12........

........ 22 34

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.

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

These are the first five terms in a Fibonacci sequence.

1 3 4 7 11

Write down the next two terms in the sequence.

In a different Fibonacci sequence the fourth term is 31 and the fifth term is 50.

Work out the first term in this sequence.

The second and third terms in the following Fibonacci sequence are x and y.

Write down algebraic expressions for the first, fourth and fifth terms.