Sequences & Series

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You might also be looking for:

• List of integer sequences
• List of OEIS sequences
• Taylor Series
• Taylor series expansion
• Expressing the function $\sin x$ as a series

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Common Sequences

Here is a short list of the common sequences.

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Further Reading:

Squares and Square Roots
(care of thirdspacelearning.com)

Squares and Square Roots Practice Questions
Triangular Numbers Practice Questions
(care of corbettmaths.com)

Squares and Square Roots - maths terminology
(care of www.bbc.co.uk/bitesize)

Squares, Cubes and Roots
(care of www.mathsgenie.co.uk)

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Arithmetic Sequences

or, algebraically,

$$U_n=a+(n-1)d$$

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Example:

Given the sequence $2,5,8,11,14,\mathrm{...}$

1. Find the next three terms of the sequence;
2. Describe the complete sequence in the form of an $n^{\mathrm{th}}$ term function;
3. Is $546$ in this sequence?

Solution

First analyse the differences in the sequence,

term number, $n$	$\mathrm{}$		$1$		$2$		$3$		$4$		$5$		$6$		$7$
term value, $U_n$	$\mathrm{}$		$2$		$5$		$8$		$11$		$\mathrm{...}$		$\mathrm{...}$		$\mathrm{...}$
difference		$\mathrm{}$		$3$		$3$		$3$		$\mathrm{...}$		$\mathrm{...}$		$\mathrm{...}$

The difference does not change between each pair of terms. This is called a common difference with the value of $3$. So, the table can be updated with the following:

term number, $n$	$0$		$1$		$2$		$3$		$4$		$5$		$6$		$7$
term value, $U_n$	$-1$		$2$		$5$		$8$		$11$		$14$		$17$		$20$
difference		$3$		$3$		$3$		$3$		$3$		$3$		$3$

This new information answers part (a) and combining with the $n^{\mathrm{th}}$ term formula gives

$$n^{\mathrm{th}}\mathit{ term}=-1+3\times \mathrm{term\; number}$$

Simplifying to

$$U_n=3n-1$$

which is the solution to part (b)

For part (c) we use the $n^{\mathrm{th}}$ term formula worked out. Equating it this to $546$

$$546=3n-1$$ $$547=3n$$ $$n=\frac{547}{3}$$ $$n=182.\stackrel{.}{3}$$

$n$ is expected to be an integer. This solution isn't, so $547$ isn't in the sequence.

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Questions and Worksheets:

Sequences: $n^{\mathrm{th}}$ term Video :

Sequences: $n^{\mathrm{th}}$ term for fractional sequences Video :

Practice Questions;

Textbook Exercise.

Worksheets.

Questions on $n^{\mathrm{th}}$ term.

(Care of corbettmaths.com)

Video on $n^{\mathrm{th}}$ term.

(Care of www.youtube.com)

Worksheets.

(Care of mathsmadeeasy.co.uk)

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Follow on:

Arithmetic Series
(Care of mmerevise.co.uk)

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Geometric Sequences & Series

A sequence is geometric if multiplying any term by a fixed non-zero number gives the next term. A geometric sequence is sometimes called a geometric progression.

If a geometric sequence has first term $a$ & a common ratio $r$ ($a,r\ne 0$), then:

• For all $n$ where $n\in \mathbb{N}$, $$r=\frac{U_{n+1}}{U_n}$$
• The terms are $a,ar,a{r}^2,a{r}^3,...$
• Each term of the sequence is given by $$U_n=a{r}^{n-1}$$

An alternating sequence is a sequence in which terms are alternately positive & negative. An alternating sequence happens when $r<0$.

A geometric sequence with a common ratio $\left|r\right|<1$ converges. This means it tends to a certain value. You call the value the limit of the sequence.

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Questions

GCSE Geometric Sequences

(Care of piacademy.co.uk)

GCSE Sequences

(Care of corbettmaths.com)

2017 Attwood G et al Pearson Edexcel A level Mathematics Pure Mathematics Year 2 Textbook London Pearson Education page 69 Ex 3c

[ MS ] (Compressed Archive)

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Geometric Series:

A Geometric series is the sum of the terms of a geometric sequence. For the geometric sequence

$$a,ar,a{r}^2,a{r}^3,...$$

the corresponding geometric series is

$$a+ar+a{r}^2+a{r}^3+...$$

The geometric series is given by

$$S_n=\frac{a(r^n-1)}{r-1}$$

or

$$S_n=\frac{a(1-r^n)}{1-r}$$

(use the second version when $r<1$ to make calculation easier)

If $\left|r\right|<1$, then the sum to infinity of the series is

$$S_{\mathrm{\infty }}=\frac{a}{1-r}$$

& the series is called a convergent series.

If $\left|r\right|>1$, then the terms of this series are getting larger, so as $n\to \mathrm{\infty }$, $S_n\to \mathrm{\infty }$. This is called a divergent series.

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Questions

A Level Edexcel Sequences and Series

(Care of www.physicsandmathstutor.com)

A Level AQA Sequences and Series

(Care of www.physicsandmathstutor.com)

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Factorial

Describing Factorial

$$n!=1\times 2\times 3\times 4\times ...\times (n-2)\times (n-1)\times n,\hspace{1em}n\in \mathbb{N}$$

First few values of $n!$

$$1!=1$$ $$2!=1\times 2=2$$ $$3!=1\times 2\times 3=6$$ $$4!=1\times 2\times 3\times 4=24$$ $$5!=1\times 2\times 3\times 4\times 5=120$$

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A value for $0!$ $$0!=1$$

Why?

One way of showing this is correct is by starting with, let's say, $8!=7!\times 8$
Then continuing this with lower values ...

$$7!=6!\times 7$$ $$6!=5!\times 6$$ $$5!=4!\times 5$$

Here we can see a pattern. So going on ...

$$4!=3!\times 4$$ $$3!=2!\times 3$$ $$2!=1!\times 2$$ $$1!=0!\times 1$$

So, $0!=1$ makes sense in this demonstration and in many equations using $0!=1$ also makes sense.
For more see Four ways $0!=1$

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Factorial Questions

Questions
(www.mathopolis.com)

Lastly, how many weeks is 10! seconds?

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Iterations

An iterative process is of method of determining the next elelment of a sequence by using previous elements

$$U_{n+1}=f(U_n,U_{n-1},...)$$

Iteration Video

Practice Questions [ MS ]

Iterative Processes Video

• iterative-methods-gcse-maths-revision-worksheets
• iteration-gcse-questions-and-worksheets
• iteration-maths

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Fibonacci Sequences

$$F_{n+2}=F_{n+1}+F_n$$ with $$F_1=a$$ and $$F_2=b$$ where $a$ and $b$ are given and often $a=b=1$

Fibonacci Sequences:

Practice Questions

Textbook Exercise

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