What is the difference between a sequence and a series ?

A sequence is an ordered list of numbers with a rule for generating them. A series is what you get by adding up the terms of a sequence.

Define periodic sequence .

A periodic sequence is one whose terms repeat in a cycle. The number of terms in each repeating cycle is called its order , or period.

Where might a periodic sequence be hiding?

In a sequence defined by a trigonometric function. Because sine and cosine repeat, a sequence built from them cycles through a small set of values.

Why is a recurrence relation not enough to define a sequence on its own?

Because it only tells you how to get from one term to the next. Without a stated first term there is nothing to start from, so the same relation could generate infinitely many different sequences.

How do you sum a sequence that is given by a recurrence relation?

If the relation is arithmetic or geometric, use the matching series formula . For any other sequence there is no general formula, so the terms are added directly or a trick has to be found that fits that particular sequence.

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General Sequences & Series (Edexcel A Level Maths: Pure): Flashcards

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What is the difference between a sequence and a series?

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What is the difference between a sequence and a series?
A sequence is an ordered list of numbers with a rule for generating them.
A series is what you get by adding up the terms of a sequence.
In sequence notation, $u_{n}$ means the $______$ term, and $S_{n}$ means the $______$ of the first $n$ terms.
In sequence notation, $u_{n}$ means the $n$ th term, and $S_{n}$ means the sum of the first $n$ terms.
So $S_{n} = u_{1} + u_{2} + u_{3} + \dots + u_{n}$ .
What makes a sequence increasing or decreasing?
Increasing means $u_{n + 1} > u_{n}$ for every positive integer $n$ , so each term is larger than the one before it.
Decreasing means $u_{n + 1} < u_{n}$ for every positive integer $n$ .
Define periodic sequence.
A periodic sequence is one whose terms repeat in a cycle.
The number of terms in each repeating cycle is called its order, or period.
True or False?
A sequence that rises for its first ten terms is an increasing sequence.
False.
The definition demands that each term beat the one before it for every positive integer $n$ , not just for the terms you happen to have checked.
A sequence can climb for a while and then turn.
Where might a periodic sequence be hiding?
In a sequence defined by a trigonometric function.
Because sine and cosine repeat, a sequence built from them cycles through a small set of values.
What does $\sum_{r = 1}^{n} u_{r}$ tell you to do?
Add up the terms $u_{r}$ , taking $r$ from $1$ up to $n$ .
The expression to the right of the $Σ$ says what is being summed, and the numbers above and below say where to start and where to stop.
True or False?
The lower limit of a sigma expression must be $1$ .
False.
It can be any integer, as in $\sum_{r = 0}^{4} (2 r + 1)$ or $\sum_{r = 7}^{11} (2 r - 13)$ .
Always read the lower limit before working out how many terms there are.
In sigma notation an arithmetic series has the form $A + B r$ , and a geometric series has the form:
$A \times B^{______}$
$A \times B^{r}$
The variable sits in the power for a geometric series, and as a multiplier for an arithmetic one.
How many terms are there in $\sum_{r = 7}^{11} (2 r - 13)$ ?
Five, since $r$ takes the values $7$ , $8$ , $9$ , $10$ and $11$ .
The count is the upper limit minus the lower limit plus one, and it is that extra one that gets forgotten.
When can two sigma expressions be combined into a single one?
When the expression being summed is the same in both and the limits join up end to end.
So $\sum_{r = 1}^{6} (4 r + 7) + \sum_{r = 7}^{11} (4 r + 7) = \sum_{r = 1}^{11} (4 r + 7)$ .
What should you watch for when a sigma expression contains more than one letter?
Only the letter written under the $Σ$ changes from term to term.
Every other letter stays fixed throughout the sum and behaves as a constant.
Define recurrence relation.
A recurrence relation gives each term of a sequence as a function of the previous term, in the form $u_{n + 1} = f (u_{n})$ .
It describes how the sequence continues rather than giving any term directly.
Why is a recurrence relation not enough to define a sequence on its own?
Because it only tells you how to get from one term to the next.
Without a stated first term there is nothing to start from, so the same relation could generate infinitely many different sequences.
An arithmetic sequence can be written $u_{n + 1} = u_{n} +______$ , and a geometric sequence as $u_{n + 1} = u_{n} \times______$ .
An arithmetic sequence can be written $u_{n + 1} = u_{n} + d$ , and a geometric sequence as $u_{n + 1} = u_{n} \times r$ .
Each also needs its first term stating, usually as $u_{1} = a$ .
True or False?
Every sequence defined by a recurrence relation is either arithmetic or geometric.
False.
A relation such as $u_{n + 1} = u_{n}^{2} - 1$ is neither, since the terms are neither added to nor multiplied by a fixed amount.
Recurrence relations reach far beyond those two families.
How do you sum a sequence that is given by a recurrence relation?
If the relation is arithmetic or geometric, use the matching series formula.
For any other sequence there is no general formula, so the terms are added directly or a trick has to be found that fits that particular sequence.

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