Mcq: Delta-f= f(x+1)-f(x); for k>1, let (Delta^k)-f = Delta((Delta^(k-1)-f)...

[Desi]

6. Let f (x) be a function of a real variable, and let \(\displaystyle \, \Delta\, f\, \) be the following function:

. . . . .\(\displaystyle \Delta\, f(x)\, =\, f(x\, +\, 1)\, -\, f(x)\)

For k > 1, put:

. . . . .\(\displaystyle \Delta^k\, f\, =\, \Delta(\Delta^{k-1}\, f)\)

Then \(\displaystyle \, \Delta^k\, f(x)\, \) equals which of the following?

. . . . .\(\displaystyle \displaystyle \mbox{A. }\, \sum_{k=0}^k\, (-1)^j\, \binom{k}{j}\, f(x\, +\, j)\)

. . . . .\(\displaystyle \displaystyle \mbox{B. }\, \sum_{k=0}^k\, (-1)^{j+1}\, \binom{k}{j}\, f(x\, +\, j)\)

. . . . .\(\displaystyle \displaystyle \mbox{C. }\, \sum_{k=0}^k\, (-1)^j\, \binom{k}{j}\, f(x\, +\, k\, -\, j)\)

. . . . .\(\displaystyle \displaystyle \mbox{D. }\, \sum_{k=0}^k\, (-1)^{j+1}\, \binom{k}{j}\, f(x\, +\, k\, -\, j)\)



Please help

 

[HallsofIvy]

You are told that

\(\displaystyle \Delta\, f(x)\, =\, f(x\, +\, 1)\, -\, f(x)\)

and that

\(\displaystyle \Delta^k\, f\, =\, \Delta(\Delta^{k-1}\, f)\)

So what are \(\displaystyle \Delta^2 f\), \(\displaystyle \Delta^3 f\), etc. Do you see a pattern?