Unimodality of the Binomial Coefficients

Pascal's triangle is a nice way to organize the binomial coefficients and its structure has led mathematicians to discover many interesting properties and identities involving the binomial coefficients. For instance, if we focus on one particular row, say row n, we see that the binomial coefficients steadily increase to a maximum value and then decrease steadily (to the first value of that row). A sequence that has this property is said to be unimodal. If we were to plot the terms of a unimodal sequence we would see that they follow a bell-like curve.

Formally, we say that a sequence a0, a1, a2, ... , an is unimodal if there exist indices i, j such that

Our goals in this lab are to prove that the sequence of binomial coefficients is unimodal and to establish bounds for the largest term in the sequence when n is even.

Exercises

    Zn Applet
  1. Use the Zn applet to the right to see that Pascal's triangle is symmetric (make sure to pick a very large modulus, say 1000, as the binomial coefficients grow very quickly). Show that
  2. Use the applet to see that each row of Pascal's triangle has a maximum (and that it is achieved at least once). What is the parity of the rows that have the maximum value occurring only once? twice?
  3. Show that if k < (n - 1)/2, then
    and that if k > (n - 1)/2, then
    What happens if k = (n - 1)/2?
  4. Show that
    (Hint: Use the binomial theorem to expand (x + 1)n and then let x = 1.) Use the above identity to show that
  5. Show that
    (Hint: What is 2n/(n+1), the average of?)
  6. Find the value of k for which
    is largest.