Prove by induction that for some natural number n: 1 + 3 + 5 +...+ (2n-1) = n^2

Question

anonymous · Answer

let p(n)=1+2+3+...+(2n-1)=n^2
prove p(1),then assume p(k)  is true 
prove p(k+1) is also true ,then by induction p(n) is also true for all n

anonymous · Answer

In case a detailed explanation is needed:

Establish the base case, for $n=1$. That is, show that
$$1=n^2$$
which should be obvious.

Now assume the equation holds for $n=k$, i.e. that
$$1+3+\cdots+(2k-1)=k^2$$
and use this to show that it holds for $n=k+1$, i.e. that
$$1+3+\cdots+(2k-1)+(2(k+1)-1)=(k+1)^2$$
As a hint, the first $k$ terms of the left side simplify to $k^2$ by the induction hypothesis. From there, it's a matter of algebraic manipulation.