If, the first number of an arithmetic sequence is 200, and the 100th number is 2, what is the 40th number of this sequence?

This problem describes an arithmetic sequence, a list of numbers where the difference between consecutive terms is always the same. To figure out the value of any term in the sequence, we first need to find this constant difference. We know the sequence travels from a starting value of 200 down to 2 over the course of 99 "steps" (the gap between the 1st and 100th term). The total change is -198 (because 2 - 200 = -198). Dividing this total change by the number of steps gives us the constant difference: -198 divided by 99 equals -2. So, each number in the sequence is simply two less than the one before it.

Once we have this rule, finding the 40th number is simple. We begin with our starting number, 200, and apply the "subtract 2" rule for the 39 steps it takes to get from the 1st term to the 40th. This calculation is 200 - (2 multiplied by 39), which equals 200 - 78. The result is 122. This type of pattern recognition is a foundational concept in mathematics, used for everything from calculating loan payments to predicting the position of moving objects.