0· A real-valued sequence is called aFibonacci sequence if it satisfies the recursion relationап+1 3 аn + an-1,for all n > 1.Let W be the set of all Fibonacci sequences.(a) Show that W is an...

o and (B")n>0, and use this to obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v. "/>
Extracted text: Let V be the vector space over R of all real-valued sequences (an) n>0· A real-valued sequence is called a Fibonacci sequence if it satisfies the recursion relation ап+1 3 аn + an-1, for all n > 1. Let W be the set of all Fibonacci sequences. (a) Show that W is an R-vector subspace of V. (b) What is the dimension of W over R? (c) Let w := 1+/5 be the golden ratio, and let ß := . Show that the sequences (@")n20 and (B")n20 form a 2 basis for W. (d) Let v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...) be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the previous two. Express v as a linear combination of the vectors (@")n>o and (B")n>0, and use this to obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.