In the series: Note 21.

Version: 0.2

By: Albert van der Sel

Doc. Number: Note 21.

For who: for beginners.

Remark: Please refresh the page to see any updates.

Status: Ready

Maybe you need to pick up "some" basic "mathematics" rather

So really..., my emphasis is on "rather

So, I am not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

The material in this text, should be more or less on the level of Highschool math

as I have observed in several European countries.

Each note in this series, is build "on top" of the preceding ones.

Please be sure that you are on a "level" at least equivalent to the contents up to, and including, note 20.

Contents:

1. Introduction Sequences.

1.1 The "Arithmetic" sequence.

1.1.1 The Recursive formula.

1.1.2 The Direct formula.

1.2 The "Geometric" sequence.

1.2.1 The Recursive formula.

1.2.2 The Direct formula.

2. The Sum formula's with sequences.

2.1 The Σ notation.

2.2 Confusion about u

2.3 The Sum formula's.

exists between those numbers.

It's not "just" a random row of numbers.

There are several "types" of sequences. The most important ones (Highschool math),

are the "Arithmetic" sequence and the "Geometric" sequence.

(2, 4, 6, 8, 10, 12....)

It does not take too much effort to see what's going on here. It looks like that

the "current term" is equal to the

And this seems to be true for any term in the sequence.

So, suppose the current term is "8" (you are currently looking at "8"). You probably agree

that "8 = 6 + 2", so here we have indeed

If you have found a relation between the "current term", and the next one (just before it,

or just after it), then that relation is called a

Here, "u

term in the row.

Note that the difference between any u

it happens to be the constant "2".

But why not choosing the relation between u

that as well. However, the convention is to write down the relation between u

Any different sequence, will have it's own recursive formula, ofcourse.

In general, a Recursive relation of these type of sequences, is of the form:

u

where C is some number (positive or negative number).

the 50th element,

You can count all the way up, until you reach that term, but we can do better than that.

Since you add "2" all the time, you only need the term you start with. The starting term of the

sequence is often denoted by u

on how you label that first term. But there are indeed a few subtleties involved.

For example, is the absolute 50th term, actually u

It depends on how you started the sequence, that is, counting from u

Or is it just that element, for which we require that u

=> If we have that sequence, labeled (starting) from u

(2, 4, 6, 8, 10, 12....) with u

This is so since we started to label as of "0". But this absolute term is not what we want.

We want to calculate u

So u

We needed to start with u

=>Suppose we have the *same* sequence, labeled (starting) from u

(2, 4, 6, 8, 10, 12....) with u

Then u

Why is this so? To make it "plausible", take a look at the numbers below:

u0 |
u1 |
u2 |
u3 |
u4 |
... |

2 |
4 |
6 |
8 |
10 |
... |

u1 |
u2 |
u3 |
u4 |
u5 |
... |

If you want to calculate "8", then:

-If starting from u

-If starting from u

But here you might say: "Forgery! you calculate u

and then try to compare the results.

No.

If you start with u

If you start with u

That's normal. Starting from the term u

way of labeling the members.

Why don't we calculate the results for n=4? Let's do it:

-starting labeling from u

-starting labeling from u

You can also see that from the table above.

Now we know what the results must be: 10 and 8. You know what the sequence is, and you can easily

verify the values from the table. Now let's calculate, using the proposed recurisve formula's:

-starting labeling from u

-starting labeling from u

The above is not exactly "proof", but hopefully, you see that in general we have:

Direct formula with u

It is very important to list a sequence, AND mention the starting label, like:

(-10, -7.5, -5, -2.5, 0, 2.5, 5...) with u

It will certainly help.

The example studied above, is an example of an

This is called this way, since the "direct formula" resembles a linear equation like f(x)=ax+b.

Just like with a linear function, if you see a

any u

This is exactly as it is, with any linear function: if you would look at discrete X'ses, like x=1, x=2, x=3 etc..,

then too we have that the difference between f(n) and f(n-1) is always constant, for any "n".

Suppose we have the sequence: (-10, -7.5, -5, -2.5, 0, 2.5, ...) with u

-Question: Give the recursive formula.

Answer: u

-Question: Give the direct formula.

Answer: u

-Question: calculate u

Answer: u

Question: suppose we have the same sequence, but now:

(-10, -7.5, -5, -2.5, 0, 2.5, ...) with u

Calculate u

Answer: u

Verify the latter answer, simply by counting as from u

You will see that you will end at "-2.5".

With recursive relations, you do not have to worry at all, with respect to the starting element.

The recursive relation simply describes the relation between any u

which is completely independent of the "starting point" of the sequence.

is a constant number, like in the first example of section 1.1: u

This time, with the "Geometric" sequence, it is

Here are two examples of Recursive relations:

u

u

The latter one, is the Recursive relation for example of the Geometric sequence:

(9, 3, 0, 1/3, 1/9, ....) with u

Just like with any type of sequence, here too we can start the sequence with the first term labeled

u

As we know, a Recursive relation is not "tied" to the choice of starting with u

as the starting term:

So, we only have one Recursive formula for a specific Geometric sequence.

u

Between the two adjacent terms, u

Between the two terms, u

Between the two term, u

etc...

It therefore seems plausible that u

But, just as with the arithmetic sequence (with a similar reasoning), we have two formats:

Direct formula with u

For example, we need to study the "sum" formula's associated with the various forms of sequences.

Let's first do a nice excercise:

Excercise:

Suppose we have the sequence (8, 4, 2, 1, 1/2, 1/4, ....) with u

-Question: what is the recursive formula?

Answer: u

-Question: what is the direct formula?

Answer: u

Let's compare "(8, 4, 2, 1, 1/2, 1/4, ....) with u

u0 |
u1 |
u2 |
u3 |
u4 |
... |

8 |
4 |
2 |
1 |
1/2 |
... |

u1 |
u2 |
u3 |
u4 |
u5 |
... |

You can easily construct the table: the next element is simply half of the former element.

So if the first element is "8", then the second is "4", the third is "2" etc..

-Question: If starting from u

Use a formula listed above:

Answer: u

-Question: If starting from u

Use a formula listed above:

Answer: u

For both types, it is true that you can sum up, say the first 7 elements, or first 10 elements,

or the first 25 elements, or any other range of elements.

Ofcourse, instead of manually adding the values yourself, there exists nifty formula's

to do it in one run. It's really simple.

-If you want to condensly (or compactly) represent the sum of 11 consequtive values, it's common

to use the Σ symbol like in:

u

Ofcourse, here you could also sum up 20, 1000 or other "n" values.

This notation just represents a summation of "n" values (from u

effected by starting the sequence from labeling as u

There is no effect, although the formula's may "look" slightly different.

You know, they will always ask something like "

or "

Take a look at the table below:

u0 |
u1 |
u2 |
u3 |
u4 |
u5 |
u6 |
u7 |
u8 |
u9 |
Here, the first 9 elements run from U0 to U8. |

u1 |
u2 |
u3 |
u4 |
u5 |
u6 |
u7 |
u8 |
u9 |
u10 |
Here, the first 9 elements run from U1 to U9. |

What we can learn here is, that if we see:

- a task to sum up the

we use Σ

- a task to sum up the

we use Σ

You can check this using the table above. You will see that the reasoning is correct.

In many cases, however, when the need for "strickt formality" is not high, showing

that "something" looks plausible, is quite enough.

Sum = Σ

-When we start labeling from u

Sum = Σ

Both formula's can be read as:

or constant decrease between adjacent elements. You can see that clearly in the recursive- or

direct formula's.

Why is it plausible? The formula "almost looks" like an average. If you look at "Sum/NumberOfTerms", then it is

really an average. But for a linear constant increase, then thats exactly

You can check that against any graph of a linear function.

Suppose we have the sequence (5, 10, 15, 20, ...) starting with u

-Question: Calculate the sum of the first 7 elements.

-Answer:

The recursive formula is: u

The direct formula is: u

Thus:

-u

-u

Note that u

Sum = ½ * 7 * (5 + 38) = 150.5

You see how easy the Sum formula works?

Sum= | u_{n}-u_{0}-------- r-1 |

-When we start labeling from u

Sum= | u_{(n+1)}-u_{1}--------- r-1 |

Here, "r" is ofcourse the factor as already was discussed in the recursive- and direct formula's, above.

Both formula's can be read as:

Sum= | NextTerm - FirstTerm ----------------- factor - 1 |

Note: the "NextTerm" really is the first "next term" in your sequence.

So, if you have the task to calculate the sum of the first 20 terms, then your

NextTerm is term 21.

Suppose we have the sequence (9, 3, 1, 1/3, 1/9, ...) starting with u

-Question: Calculate the sum for the first 7 terms.

-Answer:

The recursive formula is: u

The direct formula is: u

-u

-u

Sum= | 0.000457 - 9 --------------- -0.6666 |
= | (about) 13.5 |

Note that the last few terms are so small, that the Sum can be roughly approximated

by (-9 / -0.666) = (9 / 0.666) = about 13.5. Ofcourse, you should always calculate

up to the significant digit, as might be stated in the task description.