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Assn4Soln - 1 Explain what the linearity of(the solutions...

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1. Explain what the linearity of (the solutions of) a recurrence relation means, and prove it for the homogeneous first-order linear recurrence relations. The set of solutions of a linear recurrence relation forms a linear space. That is, if a n = f ( n ) and a n = g ( n ) are two solutions for such a recurrence, then a n = f ( n ) + g ( n ) and a n = cf ( n ) (for an arbitrary constant c ) are also solutions. A homogeneous first-order linear recurrence relation is a relation of form a n +1 = k ( n ) a n (often, we deal with the case that k ( n ) is a con- stant). If a n = f ( n ) and a n = g ( n ) satisfy this recurrence, then f ( n + 1) + g ( n + 1) = k ( n ) f ( n ) + k ( n ) g ( n ) = k ( n )( f ( n ) + g ( n )), thus a n = f ( n ) + g ( n ) also satisfies it. Similarly, we can prove that a n = cf ( n ) is also a solution. 2. Chapter 10.1, exercises 2 (in the cases (a) and (b), the solution is not unique, despite what the statement of the exercise suggests. Describe all the solutions, instead) and 6. 10.1.2 a) a n = c 1 . 5 n b) a n = c parenleftBig 5 4 parenrightBig n c) a n = 15 4 parenleftBig 4 3 parenrightBig n d) a n = 16 parenleftBig 3 2
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  • Spring '09
  • MarniMishna
  • Characteristic polynomial, Recurrence relation, Fibonacci number, first-order linear recurrence, linear recurrence relation

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