Unit 10 Section 4 : Special Sequences

Example 1

(a)

Write down the next 3 terms of the sequence,

1, 1, 2, 3, 5, 8, 13, ...

Use the first differences to extend the sequence:

...

Note that the first differences, ignoring the first 0, are in fact the actual sequence itself. These can then be used to extend the sequence:

...

(b)

Determine a formula for calculating the nth term.

Each term is the sum of the two previous terms; for example,

u_3 = u_2 + u_1

u_4 = u_3 + u_2

We can express this mathematically as,

u_n = u_n-1 + u_n-2

This formula connects u_n to the two previous terms, rather than n which we used in the earlier sections. This sequence is actually a special sequence and is called the Fibonacci sequence.

Example 2

The first two terms of a sequence are 1 and 2. The sequence is defined as,

u_n = 2u_n-1 + u_n-2

Calculate the next 3 terms of the sequence.

Note that u₁ = 1 and u₂ = 2.

u₃	= 2u_2 + u_1
	= 2 × 2 + 1
	= 5

u₄	= 2u_3 + u_2
	= 2 × 5 + 2
	= 12

u₅	= 2u_4 + u_3
	= 2 × 12 + 5
	= 29

So the first 5 terms of the sequence are,

1, 2, 5, 12, 29

Example 3

For the sequence,

6, 12, 24, 48, 96, ...

(a)

calculate the next 2 terms of the sequence,

Note that, in this sequence, each term is twice the previous term.

192

384

...

×2

(b)

determine a general formula for the nth term.

Consider how each term is formed:

u_1	= 6	= 3 × 2
u_2	= 12	= 3 × 2 × 2 = 3 × 2²
u_3	= 24	= 3 × 2 × 2 × 2 = 3 × 2³
u_4	= 48	= 3 × 2 × 2 × 2 × 2 = 3 × 2⁴

Hence the general term will be u_n = 3 × 2^n.

This sequence is an example of an exponential sequence.

Example 4

Consider the sequence,

, , , , , , ...

Determine the general formula for the nth term of the sequence.

It is best to consider the numerators and the denominators separately.

First consider the sequence of numerators,

...

As the difference between the terms is 4, we have

u_n = 4n + a

and using the first term,

3 = 4 × 1 + a

a = –1

Hence

u_n = 4n – 1

Now consider the sequence of denominators,

...

As the difference between the terms is 3, we have

u_n = 3n + b

and using the first term,

5 = 3 × 1 + b

b = 2

Hence

u_n = 3n + 2

So for the given sequence of fractions we have,

u_n =

Example 5

What happens to the sequence defined by,

u_n =

as n becomes larger and larger?

The following table lists n and u_n for several values of n.

From the table it can be seen that the values of u_n = get larger and larger as n increases.

However, the numerator is always smaller than the denominator, so each value u_n must be smaller than 1.

It follows that, as n gets larger and larger, the values of u_n must get closer and closer to 1.

n	u_n =	u_n to 3 decimal places
1	0	0
2		0.333
3		0.5
4		0.6
5		0.667
10		0.818
20		0.905
50		0.961
100		0.980
500		0.996
1000		0.998
2000		0.999

Exercises

Note: To type indeces on this page use ^ sign. e.g. n^2 :
Use braces to write formulas as indexes. e.g. 2^(n+2) :

To use multiplication sign × write * instead.

Question 1

Calculate the next three terms in each of the following sequences:

(a)	1, 3, 4, 7, 11, 18, ...	, ,
(b)	4, 9, 13, 22, 35, ...	, ,
(c)	, , , , , ...	, ,
(d)	5, 15, 45, 135, 405, ...	, ,

Question 4

Determine the formula for the general term of each of the following sequences:

(a)	15, 75, 375, 1875, 9375, ...	u_n =
(b)	1, 3, 9, 27, 81, ...	u_n =
(c)	20, 200, 2000, 20 000, 200 000, ...	u_n =
(d)	4, 28, 196, 1372, 9604, ...	u_n =

Question 9

Complete tables similar to the one in question 8, for each of the following sequences:

(a)

u_n =

n	u_n to 3 decimal places
1
5
10
50
100
500
1000
2000

As n becomes larger and larger, u_n gets closer and closer to

(b)

u_n =

n	u_n to 3 decimal places
1
5
10
50
100
500
1000
2000

As n becomes larger and larger, u_n gets closer and closer to

(c)

u_n =

n	u_n to 3 decimal places
1
5
10
50
100
500
1000
2000

As n becomes larger and larger, u_n gets closer and closer to

(d)

u_n =

n	u_n to 3 decimal places
1
5
10
50
100
500
1000
2000

As n becomes larger and larger, u_n gets closer and closer to

Question 12

Each term of a number sequence is made by adding 1 to the numerator and 2 to the denominator of the previous term.

Here is the beginning of the number sequence:

, , , , , ...

(a)

Write an expression for the nth term of the sequence.

u_n =

(b)

The first five terms of the sequence are shown on the graph.

The sequence goes on and on for ever.

Which of the following four graphs shows how the sequence continues?

Graph shows how the sequence continues.

(c)

The nth term of a different sequence is .

The first term of the sequence is .

Write down the next three terms.

, ,

(d)

This new sequence also goes on and on for ever.

Which of the four graphs below shows how the sequence continues?

Graph shows how the sequence continues.

n	u_n to 3 d.p.
1	2.000
10	3.636
100	3.960
1000	3.996

n	u_n to 3 d.p.
1	8.000
10	7.100
100	7.010
1000	7.001

n	u_n to 3 d.p.
1	4.000
10	2.105
100	2.010
1000	2.001