(a) By assuming that this canreally be done, nd coecients a; b; c; d; e that work. To
answer this question you maywant to use the Linear Algebra Toolkit (available online
- google it) or other software.Just make sure that you record the input and output
and justify what you are doing.(Hint: Substitute a few dierent values for n in this
would-be formula to generatesome linear equations in the unknowns a; b; c; d; e...Also,
to make it a bit easier foryourself, note that the rst two formulas even work for n =0,
if you interpret the sums on theleft to add up to 0. So this should also work here.)
(b)Prove that your formulareally works. (Hint: One standard way
of proving formulas like this iscalled Proof by Induction. You can learn how to do such
a proof herehttp://nrich.maths.org/4718).
Here are two famous general formulas: and For example, 1 + 2 + 3 = 6 = 4 3/2. Works! What about Let's see whether we can make an educated guess. Expanding the right sides of our first two formulas gives and So, the formula for the first sum is of the form an2 + bn + c and the formula for the second sum is of the form an3 + bn2 + cn + d (with just the right choices of coefficients). So, maybe, the next formula is of the form
