An easy way to add \( 1+2+3+4+5+6+7+8+9+10 \) is to write the sum as \( (1+10)+(2+9)+(3+8)+(4+7)+(5+6)=11+11+11+11+11=55 \). (a) Compute \( 1+2+3+\ldots+12 \) as shown above. (b) What properties of addition are you using to justify why your procedure works? (a) \( 1+2+3+\ldots+12=\square \)

To compute \( 1+2+3+\ldots+12 \) using the same pairing method, we can group the terms similarly: \[(1+12) + (2+11) + (3+10) + (4+9) + (5+8) + (6+7)\] This gives us pairs that sum to \( 13 \): \[13 + 13 + 13 + 13 + 13 + 13 = 6 \times 13 = 78\] So, \( 1+2+3+\ldots+12 = 78 \). Using this procedure, you're leveraging the commutative property of addition, which allows rearranging terms, and the associative property, which lets you combine them in groups. When you recognize that each pair of numbers adds to the same total, you confidently calculate the entire sum effortlessly!