Thread: 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?

Please Help!!

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

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 , common ratio