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

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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.all the solutions, instead) and 6....
View Full Document

This note was uploaded on 07/15/2010 for the course MACM 201 taught by Professor Marnimishna during the Spring '09 term at Simon Fraser.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online