Special sequences

The first three terms of a Fibonacci sequence are a 3 a 4 a The 5th term of this sequence is 286 Work out the value of a.

x – 4, x + 2 and 3x + 1 are three consecutive terms of an arithmetic sequence.

Find the value of x.

y – 4, y + 2 and 3y + 1 are three consecutive terms of a geometric sequence.

Find the possible values of y.

Find the value of the common ratio the sequence.

You must show all your working.

Here are the first four terms of a number sequence. 97 91 85 79

Explain how to work out the next number of the sequence.

Work out the difference between the 5th term and the 7th term of the sequence.

Explain why 52 is not a number in this sequence.

13An expression for the nth term of the sequence of triangular numbers is Prove that the sum of any two consecutive triangular numbers is a square number.

23Here are the first five terms of a geometric sequence.

Work out the next term of the sequence.

Given that the terms of this sequence are all positive,

work out the first term of this sequence. You must show all your working.

Write down the 7th odd number.

Here are the first five terms of a number sequence. 3 8 13 18 23

Write down the next two terms of this sequence.

Jim says that 50 is a term in this sequence. Jim is wrong.

Explain why.

A number sequence starts 1 2 4 Emma says that the next term is 7

Explain why Emma may be correct.

Here are the first four terms of the sequence of triangle numbers. 1 3 6 10

Find the 8th term of this sequence.

The first four terms of a Fibonacci sequence are a 2 a 3 a 5 a The sum of the first five terms of this sequence is 228 Work out the value of a.

The first four terms of a Fibonacci sequence are a 2 a 3 a 5 a The sum of the first five terms of this sequence is 228 Work out the value of a.

Show that there is only one number that is in both of these sequences.

Here are the first five terms of a sequence.

1 3 6 10 15

Write down the next two terms of the sequence.

Here are the first five terms of a Fibonacci sequence.

3 3 6 9 15

Write down the next two terms of the sequence.

The first three terms of a different Fibonacci sequence are

a a 2 a

Find the 6th term of this sequence.

S is a geometric sequence.

You must show all your working.

How many square tiles are needed to make pattern number 6?

How many circular tiles are needed to make pattern number 20?

Derek says,

“When the pattern number is odd, an odd number of square tiles is needed to make the pattern.”

Is Derek right? You must give reasons for your answer.