# set, geometric sequence,fraction and arithmetic sequence

• November 14th 2006, 12:00 AM
m777
set, geometric sequence,fraction and arithmetic sequence
please try to solve that questions.
• November 14th 2006, 03:36 AM
topsquark
Quote:

Originally Posted by m777
please try to solve that questions.

Show that the set 2 $\mathbb{Z}$ of even integers is countable.

I presume you know the theorem that if two sets are isomorphic then they have the same cardinality?

The function $f: \mathbb{Z} \to 2\mathbb{Z}: x \mapsto 2x$ is an isomorphism between the two sets as you can easily show. Thus since $\mathbb{Z}$ is countably infinite so is the set 2 $\mathbb{Z}$.

(If you don't have this theorem available, note that you can put the two sets in one to one correspondence using the above function and go from there.)

-Dan
• November 14th 2006, 03:41 AM
topsquark
Quote:

Originally Posted by m777
please try to solve that questions.

Find the 9th term of the geometric series 6, 12, 24, ...

Since it's a geometric series we know that it is of the form:
$a_{n+1} = ra_n$

or

$a_n = a_1r^{n-1}$

So $a_1 = 6$

$12 = 6 \cdot r^{2-1}$

So $r = 2$

The 9th term of the series is for n = 9:
$a_9 = 6 \cdot 2^8 = 1536$

-Dan
• November 14th 2006, 03:46 AM
topsquark
Quote:

Originally Posted by m777
please try to solve that questions.

Find the common fraction for the recurring decimal 0.7.

We may represent the decimal by the geometric series:
$\sum_{n = 1}^{\infty} 7(10)^{-n} = 0.77777....$

The sum of the terms of a geometric series $a_1r^{-n}$ is
$S = \frac{a_1}{r-1}$

In this case:
$S = \frac{7}{10-1} = \frac{7}{9}$

-Dan
• November 14th 2006, 03:51 AM
topsquark
Quote:

Originally Posted by m777
please try to solve that questions.

Which term of the arithmatic series 5, 2, -1, ... is -85?

An arithmatic series is defined as:
$a_n = a_{n-1} + k$

or

$a_n = a_1 + (n-1)k$

We know that $a_1 = 5$ and $2 = 5 - (2-1)k$. Thus $k = -3$.

$-85 = 5 + (n-1)(-3)$

$-90 = -3n + 3$

$-93 = -3n$

$n = 31$

So the 31st term will be -85.

-Dan
• November 14th 2006, 05:36 AM
Soroban
Hello, m777!

Here's alternate approach to #2 . . .

Quote:

Find the common fraction for the recurring decimal 0.7777...

We have: . $N \;= \;0.77777\hdots$

Multiply by 10: . $10N \;=\;7.77777\hdots$
Subtract $N:$ . . . . $N \;= \;0.77777\hdots$
And we have: . . $9N\:=\:7\quad\Rightarrow\quad N \,=\,\frac{7}{9}$

• November 14th 2006, 06:28 AM
ThePerfectHacker
Quote:

Originally Posted by topsquark
I presume you know the theorem that if two sets are isomorphic then they have the same cardinality?

A theorem :confused: ! You probably mean a definition.
• November 14th 2006, 11:37 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
A theorem :confused: ! You probably mean a definition.

It probably depends on the text you are using. My Topology book defines cardinality first, then introduces a proof that two sets with the same cardinality are necessarily isomorphic.

Frankly your definition might be a cleaner approach. The book rather confused me when going over this.

-Dan
• November 14th 2006, 01:10 PM
ThePerfectHacker
Quote:

Originally Posted by topsquark
[LIST][*]It probably depends on the text you are using.[/LIST My Topology book defines cardinality first,

-Dan

Never seen how to define cardinality without bijections maps.
• November 14th 2006, 02:08 PM
Plato
I would like to see that topology text.
If this is so, that text is to be avoided.
• November 14th 2006, 05:26 PM
topsquark
Quote:

Originally Posted by ThePerfectHacker
Never seen how to define cardinality without bijections maps.

Quote:

Originally Posted by Plato
I would like to see that topology text.
If this is so, that text is to be avoided.

:o I owe apologies to both of you gentlemen. The word I was remembering before the bijections was "countable" not "cardinality." Yes, in my book also it defines the bijection between sets as a definition for two sets to have the same cardinality.

-Dan
• November 15th 2006, 11:19 AM
TriKri
What's a countable set and how can an infinitly big set be countable? :confused:
• November 15th 2006, 12:51 PM
ThePerfectHacker
Quote:

Originally Posted by TriKri
What's a countable set and how can an infinitly big set be countable? :confused:

Georg Cantor developed one of the Jewels from 20th Century mathematics. The concept of sizes of sets (even infinite sets). So some infinite sets have more elements than others. Countable is any finite set, or any set which has as many elements as the number of integers. It can be shown to be the smallest type of infinite set. The Countiuum are the real numbers and they are larger. I can explain the theory but if you are not familar with the some set theory you will not follow.