<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] Sum of Arithmetic Progression $=\frac{N}{2}(a_1+a_n)$, where N is the nunber of terms
$N = \frac{(a_n-a_1)}{d} + 1 = \frac{(2n-2)}{2} + 1 = n-1+1 = n$
Thus the sum is $ = \frac{n}{2}(1+2n-1) = n^2$
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