Extending Number Sequences

The objectives of this section is to

You will have studied some sequences in earlier sections. Here we take these ideas further and meet some other types of sequences.

1

The first 4 triangular numbers are represented by the diagrams below:

(a)

Draw the next 3 triangular numbers.

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Note that an extra row of dots is added to each triangle and that the extra row has one more dot than the previous row. The next 3 triangular numbers are shown below:
(b)

Describe how to find the 8th, 9th and 10th triangular numbers without drawing the diagrams.

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To extend the sequence of triangular numbers, look at the difference between the terms:

13610152128
234567

Note that the difference between each term increases by 1 as you move along the sequence.

So,

8th term=28+8
=36
9th term=36+9
=45
10th term=45+10
=55
2

Write down the next 3 terms of the sequence:

3,   7,   10,   17,   27,   44,   71,   ...

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Look at the differences between each term:

371017274471
437101727

The first difference is not very helpful, but then note how the sequence of differences is the same as the original sequence.

For example, 10 + 7 = 17

To find the next 5 terms:

8th term=71+44
=115
9th term=115+71
=186
10th term=186+115
=301

In this type of sequence, called a Fibonacci sequence, each term is the sum of the two previous terms. For example, this sequence begins:

3, 7, 10 where 3 + 7 = 10

and the next term is 10 + 7 = 17.

Triangular Numbers1,3,6,10,15,21,28...
Square Numbers1,4,9,16,25,36,49,...
Cubic Numbers1,8,27,64,125,...
Fibonacci Sequence1,1,2,3,5,8,13,...
(formed by adding the two previous terms to get the next one)

Write down the next 4 terms of each of these sequences:

(a)
4,   7,   10,   13,   16,   19,   ,   ,   ,    
(b)
5,   11,   17,   23,   29,   35,   ,   ,   ,    
(c)
6,   8,   11,   15,   20,   26,   ,   ,   ,    
(d)
8,   10,   14,   20,   28,   38,   ,   ,   ,    
(e)
24,   23,   21,   18,   14,   9,   ,   ,   ,    
(f)
2,   12,   21,   29,   36,   42,   ,   ,   ,    
(g)
1,   1,   2,   4,   7,   11,   ,   ,   ,    

The diagram shows the first 4 square numbers:

(a)

The diagram shows the next 2 square numbers. Write their actual value underneath.


,

(b)
What is the 10th square number?
(c)
What is the 20th square number?
(d)

Find the differences between each of the first 6 square numbers in turn. What would be the difference between the 6th and 7th square numbers?

Differences = , , , , ,

Write down the next 3 terms in each of these sequences:

(i)
0,   3,   8,   15,   24,   , ,
(ii)
2,   5,   10,   17,   26,   , ,
(iii)
11,   14,   19,   26,   35,   , ,
(iv)
6,   9,   14,   21,   30,   , ,

For each sequence below, write down the number of dots in each of the first 10 diagrams:

(a)

, , , , , , , , ,
(b)

, , , , , , , , ,
(c)

, , , , , , , , ,

For each sequence below, write down the number represented by each of the first 8 diagrams:

(a)

, , , , , , ,
(b)

, , , , , , ,
(c)

, , , , , , ,

What number is represented by the 10th diagram in each of the sequences illustrated in the following diagrams:

(a)

51221
(b)

61220
(c)

81319

The Fibonacci sequence begins:

1,   1,   2,   3,   5,   8

Calculate the 10th and 20th terms in this sequence.

10th:

20th:

Write down the next 5 terms in each of these sequences:

(a)
2,   2,   4,   6,   10,   , , , ,
(b)
1,   3,   4,   7,   11,   , , , ,
(c)
2,   5,   7,   12,   19,   , , , ,
(d)
1,   9,   10,   19,   29,   , , , ,

Copy each sequence and fill in the missing terms.

(a)
,  ,  5,   9,   14,   23,   37,   ,   ,   ...
(b)
,   ,   ,   ,   20,   33,   53,   86,   139,   ...
(c)
,   ,   ,   ,   7,   11,   18,   29,   47,   ...

A sequence begins:

1,   2,   3,   6,   11,   20,   37,   68,   ...

(a)

What do you get if you add:

(i)
the first three terms:
(ii)
the 2nd, 3rd and 4th terms:
(iii)
the 3rd, 4th and 5th terms:
(b)

What are the next 3 terms in the sequence?

, ,
(c)

A similar sequence is given below. Write down the missing terms.

,   ,   ,   14,   26,   48,   88,   162,  

(d)

A sequence begins:

1,   1,   3,   5,   9,   17,   31,   ...

Write down the next 3 terms in the sequence.

, ,