3. 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 x 5 square, forming a 5 x 7 rectangle: (a) Create a sequence of rectangles using this rule starting with a 1 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 x 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 definitions for each of the sequences of perimeters you found in parts (a) and (b). Don't forget to give the initial conditions as well.
3. 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 x 5 square, forming a 5 x 7 rectangle: (a) Create a sequence of rectangles using this rule starting with a 1 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 x 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 definitions for each of the sequences of perimeters you found in parts (a) and (b). Don't forget to give the initial conditions as well.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. 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 x 5 square, forming a 5 x 7
rectangle:
5
2
(a) Create a sequence of rectangles using this rule starting with a 1 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 x 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 definitions 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)? Write down the sequences of differences and sequences
of ratios and explain anything interesting you find.
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