Homework Statement
Let f be continuous on an interval I containing 0, and define f1(x) = ∫f(t)dt, f2(x) = ∫f1(t)dt, and in general, fn(x) = ∫fn-1(t)dt for n≥2. Show that fn+1(x) = ∫[(x-t)n/n!]f(t)dt for every n≥0.
ALL INTEGRALS DEFINED FROM 0 to x (I can't format :( )
Homework...
So I have reduced the right side (a+b choose n+1) to (a+b choose n)*(a+b-n)/(n+1)
I obtained this from applying the formula I was given and applying it to both (a+b choose n) and (a+b choose n+1) and simplifying the latter to contain the former along with whatever else... in this case...
Homework Statement
Prove by induction that for any positive integers a, b, and n,
(a choose 0)(b choose n) + (a choose 1)(b choose n-1) + ... + (a choose n)(b choose 0) = (a+b choose n)
Homework Equations
(x choose y) = (x!)/((x-y)!y!)
The Attempt at a Solution
I am able to do the...