## the average molecular weights of polymers- statistics

Can someone solve this problem.....I really need the solution....I have always been bad at statistics

To create polymer molecules-linear strings of monomer units linked togehter like beads on a necklace-chemists start wit monomers in solution and intiate polymerization conditions. Monomers add to the ends of growing chains one unit at a time. Because the process is random some chaings are long, some are short. There is a distribution of molecular weights.
Suppose p is the probablility that a monomer unit is reacted and connected in the chain.
a) show that distribution of chains with length k is
p_{k}=n_{k}\sum_{k=1}^{\infty} n_{k} with $\displaystyle n_{k}=p^{k-1}(1-p)$
b) calculate the average chain length
$\displaystyle <k>=\sum_{k=1}^{\infty}kp_{k}$
in experiments one measures the molecular weight distribution $\displaystyle w_{k}=kn_{k}/\sum_{k=1}^{\infty}kn_{k}$
c) show that
$\displaystyle w_{k}=k(1-p)n_{k}$
d) compute the average molecular weight
$\displaystyle <m>=\sum_{k=1}^{\infty}kw_{k}$