Yield in terms of term In return for an investment of an amount $P$ at time $0$, dividends of amount $c$ are received at times $1, 2, \ldots, n$, respectively, and a redemption payment of $R$ is received at time $n$.
Suppose $R>P>0$ and $c>0$. Is the yield increasing or decreasing (or neither) as a function of the term $n$?
The yield equation is 
$0 = -P + c\sum_{j=1}^{n}(1+y)^{-j} + R(1+y)^{-n} \Leftrightarrow 0 = -P(1+y)^{n} + c\sum_{j=0}^{n-1}(1+y)^{j} + R = -P(1+y)^n + R + c\frac{(1+y)^n-1}{y}$
From here I can only compare for particular $n$ (say $n=0$ and $n=1$) but not in general. Also, for $P=R$ we have $y=\frac{c}{R}$, if that would help.
Any help appreciated!  
 A: When the price is below par $(P < R)$ as in this problem, the yield will, cetaris paribus, decrease as the term $n$ is increased. The behavior is the opposite if $R > P$.
A Simpler Argument
The price is given by
$$P = \sum_{j=1}^nc(1+y)^{-j} + R (1+y)^{-n}$$
Suppose we extend the term by one additional period. Maintaining the same yield $y$, the price becomes
$$\begin{align}P' &= \sum_{j=1}^{n+1}c(1+y)^{-j} + R (1+y)^{-(n+1)} \\ &= \sum_{j=1}^{n}c(1+y)^{-j} + R (1+y)^{-n}  + c(1+y)^{-(n+1)} +R(1+y)^{-n}\left((1+y)^{-1} - 1 \right) \\ &= P + (1+y)^{-(n+1)} \left(c - yR \right)\end{align}$$
If $c - yR< 0$ then $P' < P$.  We see that adding an additional period results in a lower price if the yield $y$ is held fixed. Price and yield have an inverse relationship.   In order that the price remain unchanged, the yield must be lowered. 
In this case, we have $y > c/R$. This is the condition that results in "price at a discount", $P < R.$
Original Approach
Expressing price in terms of the discounted cash flows, we see that
$$\tag{1}P = \sum_{j=1}^n \frac{c}{(1+y)^j} + \frac{R}{(1+y)^n} \leqslant nc +R$$
Using the closed form for the sum, we also have, as you have derived, 
$$P(1+y)^n = c \frac{(1+y)^n-1}{y} + R$$
Whence,
$$\tag{2}y = \left(\frac{c}{P}\frac{(1+y)^n -1 }{y} + \frac{R}{P}\right)^{1/n}-1$$
Using the binomial expansion, we have 
$$(1+y)^n = 1 + ny + \frac{n(n-1)}{2} y^2 + \mathcal{O}(y^3)$$
In typical investments, the dividend and the yield are much smaller than $1$ and a good approximation is
$$(1+y)^n \approx 1 + ny$$
Substituting into (2), we obtain 
$$\tag{3}y \approx \left(\frac{nc+R}{P}\right)^{1/n}-1$$
From (1) we have $(nc +R)/P > 1$ and it can be shown that the RHS of (3) decreases monotonically and converges to $0$ as $n \to \infty$.
