 Solution

Substitute $D_N={\beta }^N$ into the recurrence. Since $k>0$, its roots are nonzero and satisfyFor distinct roots, the general recurrence solution is $D_N=c_1{\beta }_1^N+c_2{\beta }_2^N$. The initial values require $c_1+c_2=1$ and $c_1{\beta }_1+c_2{\beta }_2=d={\beta }_1+{\beta }_2$, givingAt a repeated root, $d=\pm 2k$ and ${\beta }_1={\beta }_2=\beta =\pm k$, this quotient must be replaced by its limit:It also follows by solving the repeated-root recurrence as $(c_1+c_{2}N){\beta }^N$. This exceptional case is necessary to avoid introducing a false root in the next part; it is included in the fixed-end spring-chain determinant and its repeated-root limit.