Answer
$n^2$
Work Step by Step
We have to determine the sum:
$S=1+3+5+.....+(2n-1)$
We write each term:
$1=2\cdot 1-1$
$3=2\cdot 2-1$
$5=2\cdot 3-1$
..............................
$2n-1=2\cdot n-1$
We notice that there are $n$ terms. We add the equations side by side:
$1+3+5+...+(2n-1)=(2\cdot 1-1)+(2\cdot 2-1)+....+(2\cdot n-1)$
$S=(2\cdot 1+2\cdot 2+...+2\cdot n)-n\cdot 1$
$S=2(1+2+...+n)-n$
We use the formula:
$1+2+...+k=\dfrac{k(k+1)}{2}$
$S=2\cdot\dfrac{n(n+1)}{2}-n$
$S=n(n+1)-n$
$S=n^2+n-n$
$S=n^2$
