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5. If you were to shade in an x n square on graph paper, you could do it the boring way (with sides parallel to the edge of the paper) or the interesting way, as illustrated below: The interesting thing here, is that a 3 x 3 square now has area 13. Our goal is the find a formula for the area of a nxn (diagonal) square. a. Write out the first few terms of the sequence of areas (assume a₁ = 1, a2 = 5, etc). Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct, referring to the diagonal squares. b. Use your results from part (a) to find a closed formula for the sequence. Show your work. Note, while there are lots of ways to find a closed formula here, you should use partial sums specifically. c. Find the closed formula in as many other interesting ways as you can.

5. If you were to shade in an x n square on graph paper, you could do it the boring way (with sides parallel to the edge of the paper) or the interesting way, as illustrated below: The interesting thing here, is that a 3 x 3 square now has area 13. Our goal is the find a formula for the area of a nxn (diagonal) square. a. Write out the first few terms of the sequence of areas (assume a₁ = 1, a2 = 5, etc). Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct, referring to the diagonal squares. b. Use your results from part (a) to find a closed formula for the sequence. Show your work. Note, while there are lots of ways to find a closed formula here, you should use partial sums specifically. c. Find the closed formula in as many other interesting ways as you can.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.3: Quadratic Equations
Problem 81E
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5. If you were to shade in an x n square on graph paper, you could do it the boring way (with sides parallel to the edge of the paper) or the interesting way, as illustrated below: The interesting thing here, is that a 3 x 3 square now has area 13. Our goal is the find a formula for the area of a nxn (diagonal) square. a. Write out the first few terms of the sequence of areas (assume a₁ = 1, a2 = 5, etc). Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct, referring to the diagonal squares. b. Use your results from part (a) to find a closed formula for the sequence. Show your work. Note, while there are lots of ways to find a closed formula here, you should use partial sums specifically. c. Find the closed formula in as many other interesting ways as you can.
Transcribed Image Text:5. If you were to shade in an x n square on graph paper, you could do it the boring way (with sides parallel to the edge of the paper) or the interesting way, as illustrated below: The interesting thing here, is that a 3 x 3 square now has area 13. Our goal is the find a formula for the area of a nxn (diagonal) square. a. Write out the first few terms of the sequence of areas (assume a₁ = 1, a2 = 5, etc). Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct, referring to the diagonal squares. b. Use your results from part (a) to find a closed formula for the sequence. Show your work. Note, while there are lots of ways to find a closed formula here, you should use partial sums specifically. c. Find the closed formula in as many other interesting ways as you can.
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