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Consider the sequence defined recursively by a₁ = 1, a2 = 2, an+1 = 2an-1-lan. We can use matrix diagonalization to find an explicit formula for an an-1 a. Find a matrix that satisfies = M an M = 188 b. Find the appropriate exponent k such that an = Mk [an+1] k= D = c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. 188 P = -188 d. Find P-¹. P-¹ = an ant: e. Find M5 PD5 P-1. M5 = f. Use parts b. and e. to find a. af = g. You can find an explicit formula for an using part b. and a formula for M* = PD P-1. Try to develop this formula. Use your formula to verify the answer for part f.

Consider the sequence defined recursively by a₁ = 1, a2 = 2, an+1 = 2an-1-lan. We can use matrix diagonalization to find an explicit formula for an an-1 a. Find a matrix that satisfies = M an M = 188 b. Find the appropriate exponent k such that an = Mk [an+1] k= D = c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. 188 P = -188 d. Find P-¹. P-¹ = an ant: e. Find M5 PD5 P-1. M5 = f. Use parts b. and e. to find a. af = g. You can find an explicit formula for an using part b. and a formula for M* = PD P-1. Try to develop this formula. Use your formula to verify the answer for part f.

Advanced Engineering Mathematics
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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Consider the sequence defined recursively by a₁ = 1, az = 2, an+1 = 2an-1-lan. We can use matrix diagonalization to find an explicit formula for an an-1 = M a. Find a matrix that satisfies J an M = an+1 b. Find the appropriate exponent k such that an = Mk k = D= d. Find P-¹. c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. P = 1.88 3--188 188 P-¹ = a2 an e. Find M5 PD5 P-1. M5 = ant f. Use parts b. and e. to find a6. a6 = g. You can find an explicit formula for an using part b. and a formula for M* = PDP-1. Try to develop this formula. Use your formula to verify the answer for part f.
Transcribed Image Text:Consider the sequence defined recursively by a₁ = 1, az = 2, an+1 = 2an-1-lan. We can use matrix diagonalization to find an explicit formula for an an-1 = M a. Find a matrix that satisfies J an M = an+1 b. Find the appropriate exponent k such that an = Mk k = D= d. Find P-¹. c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. P = 1.88 3--188 188 P-¹ = a2 an e. Find M5 PD5 P-1. M5 = ant f. Use parts b. and e. to find a6. a6 = g. You can find an explicit formula for an using part b. and a formula for M* = PDP-1. Try to develop this formula. Use your formula to verify the answer for part f.
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