1. ## Geometric Sequences

In a square, one can connect the midpoints of adjacent sides to form second square. One can then connect the midpoints of adjacent sides of the second square to form a third square and continue in the same manner to form more squares successively.

Suppose the area of the initial square is 1 square unit, and consider the sequence of areas of the squares formed in this way.

Do these area form a geometric sequence? If they do, what are the first term and common ratio of the geometric sequence? If they do not, what sort of sequence do they form?

2. ## Re: Geometric Sequences

For any two successive squares in the sequence, what is the ratio of the smaller to the larger?

3. ## Re: Geometric Sequences

Hello, skweres1!

Did you make a sketch?

In a square, one can connect the midpoints of adjacent sides to form second square.
One can then connect the midpoints of adjacent sides of the second square to form a third square
and continue in the same manner to form more squares successively.

Suppose the area of the initial square is 1 square unit,
and consider the sequence of areas of the squares formed in this way.

Do these areas form a geometric sequence? . Yes!
If they do, what are the first term and common ratio of the geometric sequence?
If they do not, what sort of sequence do they form?
Code:
      *------.*.------*
|    .*:|:*.    |
|  .*:::|:::*.  |
|.*:::::|:::::*.|
*:-:-:-:+:-:-:-:*
| *:::::|:::::* |
|   *:::|:::*   |
|     *:|:*     |
*-------*-------*
We see that the inner square's area is one-half the area of the outer square.

The areas form a geometric sequence:
. . . first term $a = 1$, common ratio $r = \tfrac{1}{2}.$