Dollar-cost averaging and the harmonic mean

You invest $1,000 in a stock at the start of each of three months, when its price is $10, $20 and $40.

(a) What is your average cost per share, and how does it compare with the average of the three prices?

(b) Show that for any positive prices $p_1, \dots, p_n$, investing a fixed dollar amount each period gives an average cost per share no higher than the average price. When are they equal?

(c) Does this show that dollar-cost averaging beats investing everything up front?