<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] $$M=1+3+5+7+...+(2n-3)+(2n-1)\\M=(2n-1)+(2n-3)+...+7+5+3+1\\M+M=(1+2n-1)+(3+2n-3)+(5+2n-5)+....(2n-3+3)+(2n-1+1)\\n-term\\M+M=n(2n)\\2M=2n^2\\M=n^2
$$
</s>