Where is the error in this proof using induction?

Question

We want to prove that $\sum_{i=1}^{n} x_{i}^2 = \sum_{i=1}^{n} x_{i}t_{i}$ only if $t_i=x_i$. We use a proof by induction.

We check the first case: $\sum_{i=1}^{1} x_{i}^2 = \sum_{i=1}^{1} x_{i}t_{i}$ only if $t_i=x_i$.

We assume by inductive hypothesis that $\sum_{i=1}^{n} x_{i}^2 = \sum_{i=1}^{n} x_{i}t_{i}$ only if $t_i=x_i$.

We proceed to prove it for $n+1$.

We have that $$\sum_{i=1}^{n+1} x_{i}^2 = \sum_{i=1}^{n} x_{i}^2 + x_{n+1}^{2}$$ and by inductive hypothesis that $$\sum_{i=1}^{n} x_{i}t_{i}=\sum_{i=1}^{n} x_{i}^2$$ Therefore, $$\sum_{i=1}^{n+1} x_{i}t_{i}=\sum_{i=1}^{n} x_{i}^2 + x_{n+1}t_{n+1}$$ Then, if $\sum_{i=1}^{n+1} x_{i}^2 =\sum_{i=1}^{n+1} x_{i}t_{i}$, we can substitute and operate to have that $$\sum_{i=1}^{n} x_{i}^2 + x_{n+1}^{2}=\sum_{i=1}^{n} x_{i}^2 + x_{n+1}t_{n+1}$$ $$x_{n+1}^{2}=x_{n+1}t_{n+1}$$ $$x_{n+1}=t_{n+1}$$ As a result, we get that $\sum_{i=1}^{n+1} x_{i}^2 = \sum_{i=1}^{n+1} x_{i}t_{i}$ only if $t_i=x_i$, ending the proof.

However, the proved statement is not true. For instance, $$3^2+9^2=3*9+9*7$$

Why does the proof by induction fail?

Answer 1

Let us assume that $x_n\neq 0\;\forall\; n\in\mathbb{N}$ such that we get at least a valid base case.

The main issue is that you do not use your inductive hypothesis $$ \sum_{i=1}^n x_i^2 = \sum_{i=1}^n x_it_i \;\;\Longleftrightarrow \;\; t_i = x_i\;\forall\; i\in\{1,\ldots,n\} $$ but you use only the left side $$ \sum_{i=1}^n x_i^2 = \sum_{i=1}^n x_it_i $$ as an isolated statement. This statement is completely different from your original inductive hypothesis. You cannot derive $ \sum_{i=1}^n x_i^2 = \sum_{i=1}^n x_it_i $ from your inductive hypothesis.

Answer 2

It seems the $n=1$ case isn't even true. If $x^2=xt$, that doesn't imply that $x=t$ (particularly if $x=0$).