Why can we assume a statement is true for $n = k$, when using induction?

I know the principle of mathematical induction. The only thing that causes my confusion is that we suppose a statement is true for $n=k$ then we prove the statement is also true for $n=k+1$ but how can we suppose $n=k$ to be true? What if a statement is true for $n=k+1$ and is not true for $n=k$? Does $k$ mean to be starting from $1$ or $2$ if in the base case we prove the statement to be valid for $n=1$? Please help me with this confusion.