formulate a series expression

[lovex24]

now one of the problems our professor gave us {1,4,9,16,pi}, and we are supposed to generate a formula so that when plugging in 1, 2, 3, 4, 5, the function would yield 1, 4, 9, 16, pi respectively as results.

I did come up with something, but when I punch that into the calculator, for some reason it doesn't generate a value at 5, and I can't find out why because when I do it, i.e. I plug in the value into the function myself and calculate by hand, the result does match pi. My assumption is that it's either I made a mistake while calculating or the calculator can't generate a visible value if there is excessive part of discontinuity(I doubt it). I'd really appreciate if someone can help me out. Technically this was a problem given in our Calc 2 class, but I felt like this relates more to algebra that's why I posted it here.

 

[lovex24]

You keep posting problems that make no sense.

PLEASE post the original problem exactly as it appears in your book or as given to you.

no offense but I don't understand how this problem(or any other problem I've posted before for that matter) makes no sense. I even explained what I am supposed to do. OK since you really want to know, it was a story my professor shared with us. A guy goes to interview, but apparently he was nervous. The employer asks him if there's a set of numbers {1, 4, 9, 16}, what's the next number going to be? Obviously the employer is expecting him to answer 25, but since he was so nervous(at least that's what our professor told us) he responded with PI instead. Now the question is, hypothetically speaking, if the set of numbers really are {1, 4, 9, 16, pi}, what general formula does it need to have in order for the input {1, 2, 3, 4, 5}to have the said output? I mean whether the story was true or not(most likely not), she assigned it as a challenge problem and I figured I'd give it a shot. I apologize if I haven't made it clear myself, but somehow I felt I did. Now if you still don't understand the problem it really is not my fault.

 

[lookagain]

now one of the problems our professor gave us {1,4,9,16,pi},
and we are supposed to generate a formula so that when plugging in 1, 2, 3, 4, 5,
the function would yield 1, 4, 9, 16, pi respectively as results.

I don't know how to do it by one of the more elegant ways, so I set up 5 equations with

5 unknowns (a, b, c, d, and e) and solved them simultaneously.



\(\displaystyle f(x) \ = \ ax^4 + bx^3 + cx^2 + dx + e\)

The first equation is \(\displaystyle \ \ 1 = a + b + c + d + e \)

The second equation is \(\displaystyle \ \ 4 = 16a + 8b + 4c + 2d + e \)



And so on to the fifth equation.



After solving for a, b, c, d, and e, exactly, the function is


\(\displaystyle f(x) \ = \ \bigg(\dfrac{\pi - 25}{24}\bigg)x^4 \ + \ \bigg(\dfrac{125 - 5\pi}{12}\bigg)x^3 \ + \ \bigg(\dfrac{35\pi - 851}{24}\bigg)x^2 \ + \ \bigg(\dfrac{1250 - 50\pi}{24}\bigg)x \ + \ (\pi - 25)\)



For example, using this formula I verified that \(\displaystyle \ f(1) \ = \ 1.\)


Perhaps some others may want to verify if the remaining cases are true:


\(\displaystyle f(2) \ = \ 4\)

\(\displaystyle f(3) \ = \ 9\)

\(\displaystyle f(4) \ = \ 16\)

\(\displaystyle f(5) \ = \ \pi\)

 

[Bob Brown MSEE]

Lagrange Interpolating Polynomial

http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html