Problem:Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.)
Proof: Base Case: Let there be three vectors namely v1,v2,v3. We see that (v1+v2)+v3=v1+(v2+v3).
Inductive Case: Let k be the number of vectors where k∈N. We want to assume that for v1...vkthat we could associate them any two ways and get the same sum. We want to show for k+1 vectors that the same thing holds.
Case 1: Odd number of k+1 vectors
( (v1+v2)+...+(vk-1+vk))+vk+1 =(v1+ (v2+v3)+...+(vk+Vk+1))=v1+ ((v2+v3)+...+(vk+Vk+1))
Case 2: Even Number of k+1 vectors
Hence by induction it holds for k+1 vectors.
Did I do it correctly?