5. We define a sequence an by an = an-1-an_2+an-3+2(2n 3) for n 4...

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5. We define a sequence an by an = an-1-an_2+an-3+2(2n – 3) for n ≥ 4 and with initial terms a₁ = 1, a2 = 4, and a3 = 9. (a) Calculate a4, a5, a6, and a7 by hand. (b) Calculate the first 100 terms using a computer (eg. spreadsheet or script) and plot the values on an xy-plane. Include the image using the includegraphics command in LaTeX. Be sure the image is of reasonable size in your PDF. (c) Based on your experimentation/observations above, conjecture a closed-form expression for the se- quence and briefly explain how you arrived at your conjecture. (d) Prove by induction the correctness of your closed-form expression. 5. We define a sequence an by an = an-1-an_2+an-3+2(2n – 3) for n ≥ 4 and with initial terms a₁ = 1, a2 = 4, and a3 = 9. (a) Calculate a4, a5, a6, and a7 by hand. (b) Calculate the first 100 terms using a computer (eg. spreadsheet or script) and plot the values on an xy-plane. Include the image using the includegraphics command in LaTeX. Be sure the image is of reasonable size in your PDF. (c) Based on your experimentation/observations above, conjecture a closed-form expression for the se- quence and briefly explain how you arrived at your conjecture. (d) Prove by induction the correctness of your closed-form expression.

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a To calculate a we substitute n 4 into the given formula a a a a 224 3 a a a 28 3 9 4 1 25 9 4 1 10 16 Similarly we can calculate a a and a a a a a 2... View the full answer