Calvin's Christmas Candy Canes

Solution 1: He ate 1 candy cane on the first day, 3 candy canes on the second day, 5 candy canes on the third day, so on and so forth till 23 candy canes on the 12th day. Let the sum be denoted by $S$ . Then using Pascal's trick, we have

$\begin{array}{llllll} S & = 1 & +3 & + 5 & + \ldots & + 23 \\ S & = 23 &+ 21 &+ 19 & + \ldots & + 1 \\ \hline 2S & = 24 & + 24 & + 24 & + \ldots & + 24 \\ \end{array}$

Hence, $2S = 24 \times 12 = 288$ , so $S = \frac {288}{2} = 144$ .

Solution 2: This is an arithmetic progression with initial term 1, common difference 2, and number of terms 12. Hence, the sum of all terms is $\frac {1 + (1+ 11 \times 2) } {2} \times 12 = 144$ .