Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.1 Sequences - 11.1 Assess Your Understanding - Page 827: 56

$1+3+5+7+...+(2n-1)$

We see that there are $n$ terms, as the index $k$ changes from $0$ to $n$. The index $k$ indicates how the terms are formed. We write out the sum for the terms as follows: $\displaystyle \sum_{k=0}^{n-1}(2k+1)=\\=[2(0)+1]+[2(1)+1]+[2(3)+1]+...+[2(n-1)+1] \\=1+3+5+7+...+(2n-1)$