discret math Arithmetic and Geometric Sequences 12. Starting with any rectangle, we can create a new, larger rectangle byattaching a square to the longer side. For example, if we start with a 2 x 5rectangle, we would glue on a 5 × 5 square, forming a 5 x 7 rectangle:5527The next rectangle would be formed by attaching a 7 x 7 square to the top orbottom of the 5 × 7 rectangle.a. Create a sequence of rectangles using this rule starting with a1 x 2rectangle. Then write out the sequence of perimeters for the rectangles(the first term of the sequence would be 6, since the perimeter of a 1 × 2rectangle is 6 - the next term would be 10).b. Repeat the above part this time starting with a 1 x 3 rectangle.c. Find recursive formulas for each of the sequences of perimeters youfound in parts (a) and (b). Don't forget to give the initial conditions as well.d. Are the sequences arithmetic? Geometric? If not, are they close to beingeither of these (i.e., are the differences or ratios almost constant)? Explain.

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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discret math
Arithmetic and Geometric Sequences

12. Starting with any rectangle, we can create a new, larger rectangle by
attaching a square to the longer side. For example, if we start with a 2 x 5
rectangle, we would glue on a 5 × 5 square, forming a 5 x 7 rectangle:
5
5
2
7
The next rectangle would be formed by attaching a 7 x 7 square to the top or
bottom of the 5 × 7 rectangle.
a. Create a sequence of rectangles using this rule starting with a1 x 2
rectangle. Then write out the sequence of perimeters for the rectangles
(the first term of the sequence would be 6, since the perimeter of a 1 × 2
rectangle is 6 - the next term would be 10).
b. Repeat the above part this time starting with a 1 x 3 rectangle.
c. Find recursive formulas for each of the sequences of perimeters you
found in parts (a) and (b). Don't forget to give the initial conditions as well.
d. Are the sequences arithmetic? Geometric? If not, are they close to being
either of these (i.e., are the differences or ratios almost constant)? Explain.

Expert Solution

How the new rectangles formed from previous rectangles ?

In each step a square is added to previous rectangle to get rectangle.The main question is how we decide what is the size of square and which way it should be added .

we have to add the maximum size of square such that the new figure is also a rectangle .Let us suppose the size of rectangle is $m \times n$ where $m < n$ then add square of length $n$ .It has to be added such that the one side common to both square and old rectangle.

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