Math  /  Algebra

QuestionConsider the sequence defined recursively by a1=1,a2=1,an+1=10an1+7ana_{1}=-1, a_{2}=1, a_{n+1}=-10 a_{n-1}+7 a_{n}. We can use matrix diagonalization to find an explicit formula for ana_{n}. a. Find a matrix that satisfies [anan+1]=M[an1an]\left[\begin{array}{c}a_{n} \\ a_{n+1}\end{array}\right]=M\left[\begin{array}{c}a_{n-1} \\ a_{n}\end{array}\right] b. Find the appropriate exponent kk such that [anan+1]=Mk[a1a2]k= 媌 \begin{array}{l} {\left[\begin{array}{c} a_{n} \\ a_{n+1} \end{array}\right]=M^{k}\left[\begin{array}{l} a_{1} \\ a_{2} \end{array}\right]} \\ k=\square \text { 媌 } \end{array} c. Find a diagonal matrix DD and an invertible matrix PP such that M=PDP1M=P D P^{-1}. d. Find P1P^{-1}. e. Find M5=PD5P1M^{5}=P D^{5} P^{-1}. f. Use parts b. and e. to find a6a_{6}. a6=a_{6}= \square g. Develop an explicit formula for ana_{n} using part b. and a formula for Mk=PDkP1M^{k}=P D^{k} P^{-1}.

Studdy Solution
Develop an explicit formula for an a_n using: an=[10]Mn1[a1a2]a_n = \begin{bmatrix} 1 & 0 \end{bmatrix} M^{n-1} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}
Substitute Mn1=PDn1P1 M^{n-1} = P D^{n-1} P^{-1} to express an a_n in terms of the diagonalized form.
The explicit formula for an a_n is derived using the diagonalization approach.

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