Question 1013712: You are given a multiplication table, to make your first “move” you pick any space on the table. To make your second move, you pick any negative “slope” and move in the table using that slope. For example, in the table below the person chose -2 as their slope and 1x1 as their starting point. Find a formula for the nth move in the above situation. Explain how you arrived at your answer.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! I'm guessing your example would go from 1 --> 6 --> 15 --> 28 --> 45, etc. Is this correct? (I am using a multiplication table given by https://en.wikipedia.org/wiki/Multiplication_table, under "History")
Assuming this is true, first off, the negative slope would have to be rational (for obvious reasons). Furthermore, the rational number would probably need to be in lowest terms, as -2 and -4/2 are different here.
If we denote the starting point by r*c, and the "slope" by -a/b where a and b are relatively prime positive integers, then the second element is (r+b)(c+a). The third element is (r+2b)(c+2a). In general, the nth element is (r+(n-1)b)*(c+(n-1)a).
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