2015 AMC 10A Problem 3

Let us try to find a pattern between the number of toothpicks needed for the staircases. For a $1$-step staircase, we would only need $4$ toothpicks (just a square). For a $2$-step staircase, we would need $10$ toothpicks according to the diagram. Similarly, we would need $18$ toothpicks for a $3$-step staircase. A $2$-step staircase needs $10 - 4 = 6$ more toothpicks than a $1$-step staircase. A $3$-step staircase needs $18 - 10 = 8$ more toothpicks than a $2$-step staircase. Following this pattern, we can see that a $4$-step staircase will need $18 + 10 = 28$ toothpicks, and a $5$-step staircase will need $28 + 12 = 40$ toothpicks. This means that Ann would need to add $40 - 18 = 22$ more toothpicks. Thus, D is the correct answer.