Question Appendix.1: Sum of Odd Integers Use the pattern to propose a formula for...
Sum of Odd Integers
Use the pattern to propose a formula for the sum of the first n odd integers.
\begin{matrix}\quad\quad\quad\quad\quad\quad\quad 1 = 1 \\\quad\quad\quad\quad\quad 1 + 3 = 4\\\quad\quad\quad 1 + 3 + 5 = 9\\\quad\quad 1 + 3+ 5+ 7 = 16\\ 1 + 3 + 5 + 7 + 9 = 25\end{matrix}Notice that the sums on the right are equal to the squares 1^2, 2^2, 3^3, 4^4, and 5^5. Judging from this pattern, it appears that the sum S_n of the first n odd integers is
S_n = 1 + 3 + 5+ 7 + \cdot \cdot \cdot + (2n – 1)= n^2.
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