Final answer:
Utilizing inductive reasoning, the next line in the pattern based on mathematical sequences is predicted to be 17 * 18 = 19 * 20 - (17 + 18 + 19 + 20).
Step-by-step explanation:
Inductive reasoning in mathematics allows us to identify patterns and make predictions. Given the pattern 15 * 16 = 17 * 18 - (15 + 16 + 17 + 18); 16 * 17 = 18 * 19 - (16 + 17 + 18 + 19), let's apply inductive reasoning to predict the next line in this sequence.
First, let's observe the operations taking place. In each equation, one side multiplies two consecutive integers, while the other side multiplies the next two consecutive integers and then subtracts the sum of both pairs of consecutive integers. For instance, in the first example, 15 and 16 are multiplied, and then 17 and 18 are multiplied with the sum (15 + 16 + 17 + 18) subtracted from this product. The same pattern holds for the second example with consecutive integers 16, 17, 18, and 19.
Noticing this pattern, let's predict the next line in the sequence:
- Multiply two consecutive integers starting after the last integer used in the previous line, which is 17.
- The next consecutive integers to be used are 17 and 18.
- According to the pattern, we should multiply the next two integers, which are 19 and 20.
- Finally, subtract the sum of the four consecutive integers (17 + 18 + 19 + 20) from the product of the last two numbers.
Applying these steps:
17 * 18 = 19 * 20 - (17 + 18 + 19 + 20)
According to inductive reasoning, we predict the above equation to be the next in the pattern. Through this example, we can appreciate that mathematics provides many paths to the same answer, and inductive reasoning is a useful tool in the process of discovering these paths.