  We discuss and compare upper and lower bounds obtained by two different
methods for the positive zero of the ultraspherical polynomial
$C_{n}^{(\lambda)}$ that is greater than $1$ when $-3/2 < \lambda < -1/2.$ Our
first approach uses mixed three term recurrence relations and interlacing of
zeros while the second approach uses a method going back to Euler and Rayleigh
and already applied to Bessel functions and Laguerre and $q$-Laguerre
polynomials. We use the bounds obtained by the second method to simplify the
proof of the interlacing of the zeros of $(1-x^2)C_{n}^{(\lambda)}$ and
$C_{n+1}^{(\lambda)}$, for $-3/2 < \lambda < \infty$.
