Sequences and Series

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Number Patterns[edit]

An important skill in mathematics is to be able to:

  • recognise patters in sets of numbers,
  • describe the patters in words, and
  • continue the pattern

A list of numbers where there is a pater is called a number sequence. The members (numbers) of a sequence are said to be its terms.

Example[edit]


3, 7, 11, 15, ...

The above is a type of number sequence. The first term is 3, the second is 7, etc. The rule of the sequence is that "the sequence starts at 3 and each term is 4 more than the previous term."

Arithmetic Sequences[edit]

An arithmetic sequence is a sequence in which each term differs from the previous by the same fixed number:


2, 5, 8, 11, 14, ...
is arithmetic as 5-2=8-5=11-8=14-11
etc

Algebraic Definition[edit]

Within an arithmetic sequence, the nth term is defined as follows:


u_n = u_1 + (n-1)d

Where d is defined as:


d = u_{n+1} - u_n


Here, the notation is as follows:

u_1 is the first term of the sequence.
n is the number of terms in the sequence.
d is the common difference between terms in an arithmetic sequence.

Example[edit]

Given the sequence 1, 3, 5, 7 ... n, the values of the notation are as follows:


d = u_{n+1} - u_n


d = u_2 - u_1 = 3 - 1


d = 2

And


u_1 = 1

Therefore


u_n = u_1 + (n-1)d


u_n = 1 + (n-1)2


u_n = 1 + 2n - 2


u_n = 2n - 1

Thus we can determine any value within a sequence:


u_5 = 2(5) - 1 = 10 - 1 = 9

Arithmetic Series[edit]

An arithmetic series is the addition of successive terms of an arithmetic sequence.


21 + 23 + 27 + ... + 49

Sum of an Arithmetic Series[edit]

Recall that if the first term is u_1 and the common difference is d, then the terms are:


u_1, u_1 + d, u_1 + 2d, u_1 + 3d, ...

Suppose that u_n is the last or final term of an arithmetic series. Then, where S_n is the sum of the arithmetic series:


S_n = u_1 + (u_1 + d) + (u_1 + 2d) + ... + (u_n - 2d) + (u_n - d) + u_n

However, if one were to reverse the series, like so:


S_n = u_n + (u_n - d) + (u_n - 2d) + ... + (u_1 + 2d) + (u_1 + d) + u_1

Then add the two sequences together:


2S_n = (u_1+u_n) + (u_1 + u_n) + (u_1 + u_n) + ... + (u_1 + u_n) + (u_1 + u_n) + (u_1 + u_n)

One can see that there there in fact n terms that look identical, thus:


2S_n = n(u_1 + u_n)


S_n = \frac{n}{2}(u_1 + u_n)

Geometric Sequences[edit]

A sequence is geometric if each term can be obtained from the previous one by multiplying by the same non-zero constant.


2, 10, 50, 250, ...
is geometric as 
2 \times 5 = 10
and 
10 \times 5 = 50
and 
50 \times 5 = 250

Notice that


\frac{10}{2} = \frac{50}{10} = \frac{250}{50} = 5
i.e., each term divided by the previous one is a non-zero constant.

Algebraic definition[edit]


\{u_n\}
is geometric 
\Leftrightarrow \frac{u_{n+1}}{u_n} = r
for all positive integers 
n
wherer is a constant (the common ratio)

The 'Geometric' Mean[edit]

If a,bandc are any consecutive terms of a geometric sequence then


\frac{b}{a} = \frac{c}{b}
{equating common ratios}

Therefore


b^2 = ac
and so 
b=\pm\sqrt{ac}
where 
\sqrt{ac}
is the geometric mean of 
a
and 
c
.

The General Term[edit]

Suppose the first term of a geometric sequence is u_1 and the common ratio isr.

Then u_2 = u_1rthereforeu_3=u_1r^2etc.

Thusu_n=u_1r^{n-1}

u_1 is the first term of the sequence.
n is the general term
r is the common ratio between terms in an geometric sequence.

Geometric Series[edit]

Compound Interest[edit]

Compound interest is the interest earned on top of the original investment. The interest is added to the amount. Thus the investment grows by a large amount each time period.

Consider the following

You invest $1000 into a bank. You leave the money in the back for 3 years. You are paid an interest rate of 10% per annum (p.a.). The interest is added to your investment each year.

An interest rate of 10% p.a. is paid, increasing the value of your investment yearly.

Your percentage increase each year is 10%, i.e.,


100% + 10% = 110%

So 110% of the value at the start of the year, which corresponds to a multiplier of 1.1.

After one year your investment is worth


$1000 \times 1.1 = $1100

After two years it is worth After three years it is worth
$1100 \times 1.1 $1210 \times 1.1
= $1000 \times 1.1 \times 1.1 = $1000 \times (1.1)^2 \times 1.1
= $1000 \times (1.1)^2 = $1000 \times (1.1)^3
= $1210 = $1331

Note

u_1 = $1000 = The initial investment
u_2 = u_1\times1.1 = The amount after 2 year
u_3 = u_1\times(1.1)^2 = The amount after 3 years
u_4 = u_1\times(1.1)^3 = The amount after 4 years
\vdots
u_{n+1} = u_1 \times (1.1)^n = amount after n years

In general, u_{n+1}=u_1 \times r^nis used for compound growth, where

u_1 is the initial investment
r is the growth multiplier
n is the number of years
u_{n+1} is the amount after n years