{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Assn4Soln

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.
• Spring '09
• MarniMishna
• Characteristic polynomial, Recurrence relation, Fibonacci number, first-order linear recurrence, linear recurrence relation

{[ snackBarMessage ]}