The rectangle shows an array of nine numbers represented by combinations of the variables a, b, and c.
a + b | a − b − c | a + c |
a − b + c | a | a + b − c |
a − c | a + b + c | a − b |
a. Determine the nine numbers in the array for a = 10, b = 6, and c = 1. What do you observe about the sum of the numbers in all rows, all columns, and the two diagonals?
b. Repeat part (a) for a = 12, b = 5, and c = 2.
c. Repeat part (a) for values of a, b, and c of your choice.
d. Use the results of parts (a) through (c) to make an inductive conjecture about the rectangular array of nine numbers represented by a, b, and c.
e. Use deductive reasoning to prove your conjecture in part (d).
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