2, 3, and challenge activity

Exploration 1.4.2: Triangular Different f(n) = ank Take a look at the sequence of square numbers listed in the first row of numbers below. The first several terms of the sequence are listed and, below that, the first differences of the two numbers immediately above, and below that are the differences of the differences (i.e., the second differences). 81 ... sequence = 1 49 - 64 25. 36 15 17 ... 13 udifference- Notice that for the sequence n the second differences are all 2. 1. Now explore the differences for the sequence of cubes n3 . Do you notice any patterns? whatever the exponent 13 - 1, 8, 27.... valve is, is the difference level Where all # are the same 2. Make a conjecture as to what will happen with the fourth powers and the fifth powers. Make a conjecture about the differences and the resulting constant for the sequence of &:th powers of nk where k is a positive integer. [This Exploration can be done by hand, or with a calculator, or in a spreadsheet program such as Excel.] 14-1, 16, 81 .... 3. If you are given a sequence of kth powers ank, is there a way to figure out the value of the coefficient a? y = ank find K & A 4. Explain how the triangular differences process is related to the rate of change of your sequence function. As a challenge activity, you might consider sequences generated by explicit equations containing multiple terms of the form an*. For example, try applying triangular differences for the sequence generated by n2, n, and n2 - 3n. Do you notice any patterns? What about 2n2 + 4n or 5n2 + 2n - 5? This exercise is most efficiently investigated using a computer spreadsheet program. 32