# Thread: Mixed Arithmetic/Geometric Progression and Infinite Algebra

1. ## Mixed Arithmetic/Geometric Progression and Infinite Algebra

I would just like to consult these two problems that I encountered in my textbook. I am quite frustrated since i can't figure both of these questions as I was reviewing for my exams. If you are kind enough to help and guide me, I'd thank you with all my heart. (I am having a feeling that a kind of question like this will appear in my exam on tuesday)

Question 1:

Let X1, X2, X3, .... Xk be an Arithmetic Sequence and Y1, Y2, Y3, ...Ykbe a Geometric Sequence.
Define the sequence Z1, Z2, Z3, ....Zkby Zk = Xk + Yk for every K element of Natural numbers.
If Z1=1, Z2=8, Z3=32, Find Z5.

Question 2:

Let S0 be a square whose side has length of 1 unit.
Let S1 be the square whose vertices are the midpoints of the sides of S0. (connecting the midpoint of consecutive sides)
Let S2 be the square whose vertices are the midpoints of the sides of S1, and so on.
We do this infinitely many times to obtain squares Sk where K is a Natural number.
If ASk denotes the AREA of square Sk, Find AS1 + AS2 + AS3 + ... + ASk.

2. Hello, potatocount!
Question 2:

Let S0 be a square whose side has length of 1 unit.
Let S1 be the square whose vertices are the midpoints of the sides of S0,
. .
(connecting the midpoint of consecutive sides)
Let S2 be the square whose vertices are the midpoints of the sides of S1, and so on.
We do this infinitely many times to obtain squares Sk where K is a Natural number.

If $A_k$ denotes the area of square $S_k$, find: .$A_1 + A_2 + A_3 + \cdots + A_k$

I note that $A_0$ is omitted from the sum
. . and assume that this is deliberate.

Code:
      *-------*-------*
|     *   *     |
|   *-------*   |
| * |       | * |
1 *   |       |   *
| * |       | * |
|   *-------*   |
|     *   *     |
*-------*-------*
1

The side of $S_0$ is $1$; its area is $A_0 \:=\: 1^2 \:=\: 1$

The side of $S_1$ is $\frac{1}{\sqrt{2}}$; its area is $A_1 \:=\: \left(\frac{1}{\sqrt{2}}\right)^2 \:=\:\frac{1}{2}$

The side of $S_2$ is $\frac{1}{2}$; its area is $A_2 \:=\:\left(\frac{1}{2}\right)^2 \:=\:\frac{1}{4}$

We find that the area of a square is one-half that of the preceding square.

The desired total is: .$T_k \:=\:\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^k}$

This is a geometric series with first term $a=\frac{1}{2}$ and common ratio $r = \frac{1}{2}$
. . Its sum is: .$a\,\frac{1-r^k}{1-r}$

Therefore: .$T_k \;=\;\frac{1}{2}\cdot\frac{1-\left(\frac{1}{2}\right)^k}{1-\frac{1}{2}} \;=\;1 - \left(\frac{1}{2}\right)^k$

3. Originally Posted by soroban

I note that $A_0$ is omitted from the sum
. . and assume that this is deliberate.

Code:
      *-------*-------*
|     *   *     |
|   *-------*   |
| * |       | * |
1 *   |       |   *
| * |       | * |
|   *-------*   |
|     *   *     |
*-------*-------*
1

Hello SOROBAN!

Thank you for taking your time answering my question, I would just like to ask what would be the answer if A0 is included in the calculation of Ak?
Wouldn't this problem be using an "infinite sum" formula? Sinfinity = A1 / (1-r) ?

And if it is not too much to ask, any good news on Question#1? thanks!

4. Originally Posted by potatocount
. > > > If you are kind enough to help and guide me, < < < ...Question 1:

Let X1, X2, X3, .... Xk be an Arithmetic Sequence and Y1, Y2, Y3, ...Ykbe a Geometric Sequence.
Define the sequence Z1, Z2, Z3, ....Zkby Zk = Xk + Yk for every K element of Natural numbers.
If Z1=1, Z2=8, Z3=32, Find Z5.

> > > potatocount, you have shown no work at all for you to be helped and guided.
And a worked-out problem by anyone here for you is not "helping and guiding" you.

Question 2:

Let S0 be a square whose side has length of 1 unit.
Let S1 be the square whose vertices are the midpoints of the sides of S0.
(connecting the midpoint of consecutive sides)
Let S2 be the square whose vertices are the midpoints of the sides of S1, and so on.
We do this infinitely many times to obtain squares Sk where K is a Natural number.
If ASk denotes the AREA of square Sk, Find AS1 + AS2 + AS3 + ... + ASk.

> > > Again, potatocount, you showed no work. This is in reference to this second problem.

[QUOTE=potatocount;316892]
Hello SOROBAN!

1) > > >
[QUOTE]
Originally Posted by potatocount

2) > > >
And if it is not too much to ask, any good news on Question#1? thanks!

potatocount,

"help and guidance."

2) Again and again, do not ask for "any good news" about a certain question
when haven't contributed any work yourself.