Consider the sequence defined recursively by a1 = -1. 02-2, an+1=-8an-1-9an We can use matrix diagonalization to find an explicit formula for an a. Find a matrix that satisfies M= D= 0 p-1= -9 b. Find the appropriate exponent & such that an - Mk Gn+1 k-n-1 c. Find a diagonal matrix D and an invertible matrix P such that M PDP-¹, -1 3--12 0 d. Find P-1, -8 M- 4/3 0₂ -8 1/6 -1/3 e. Find MPD P-1, an an+1 -1/6 = M an -8 f. Use parts b. and e. to find a d g. You can find an explicit formula for a, using part b. and a formula for MPDP-1. Try to develop this formula. Use your formula to verify the answer for part f

Consider the sequence defined recursively by a1 = -1. 02-2, an+1=-8an-1-9an We can use matrix diagonalization to find an explicit formula for an a. Find a matrix that satisfies M= D= 0 p-1= -9 b. Find the appropriate exponent & such that an - Mk Gn+1 k-n-1 c. Find a diagonal matrix D and an invertible matrix P such that M PDP-¹, -1 3--12 0 d. Find P-1, -8 M- 4/3 0₂ -8 1/6 -1/3 e. Find MPD P-1, an an+1 -1/6 = M an -8 f. Use parts b. and e. to find a d g. You can find an explicit formula for a, using part b. and a formula for MPDP-1. Try to develop this formula. Use your formula to verify the answer for part f

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.2: Sequences, Series And Summation Notation

Problem 42E

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