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 namelyv. We see that_{1},v_{2},v_{3}(v_{1+}v_{2})_{+}v_{3=}vInductive Case: Let k be the number of vectors where k∈N. We want to assume that for_{1+}(v_{2}_{+}v_{3}).

vthat 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._{1...}v_{k}

Case 1: Odd number of k+1 vectors

((v_{1+}v_{2})+...+(v_{k-1}+v_{k}))+v_{k+1 }=(v_{1+}(v_{2}_{+}v_{3})+...+(v_{k}+V_{k+1}))=v1+((v_{2}_{+}v_{3})+...+(vCase 2: Even Number of k+1 vectors_{k}+V_{k+1}))

((v_{1+}v_{2})+...+(v_{k-2}+v_{k-1})+v_{k})+v_{k+1}=(v_{1}+(v_{2}+v_{3})+..+(v_{k-1}+v_{k}))+v_{k+1}=(v_{1}+v_{2})+...+(v_{k}+v_{k+1})=v_{1}+((v_{2}+v_{3})+..+(v_{k}+v_{k+1}))

Hence by induction it holds for k+1 vectors.

Did I do it correctly?