<s>[INST] Closed form for $1 + 3 + 5 + \cdots +(2n-1)$ What is the closed summation form for $1 + 3 + 5 + \cdots + (2n-1)$ ?
I know that the closed form for $1 + 2 + 3+\cdots  + n = n(n+1)/2$ and I tried plugging in $(2n-1)$ for $n$ in that expression, but it didn't produce a correct result:
$(2n-1)((2n-1)+1)/2$
plug in 3 
$(2n-1)((2n-1)+1)/2 = 6 != 1 + 2 + 3 = 9$
 [/INST] Let $$S_n=1+2+3+...+(2n-1)$$
Clearly
$S_1=1=1^2$
$S_2=1+3=2^2$
$S_3=1+3+5=9=3^2$
Suppose $S_n=n^2,$ then $S_{n+1}=S_n+(2n+1)=(n+1)^2$
By mathematical induction $$S_n=n^2 ,\forall n\in \mathbb{N}$$
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