[Algebra]
Problem: Let $T=55$. Compute the value of $K$ such that $20, T-5, K$ is an increasing geometric sequence and $19, K, 4 T+11$ is an increasing arithmetic sequence.

Solution: The condition that $20, T-5, K$ is an increasing geometric sequence implies that $\frac{T-5}{20}=\frac{K}{T-5}$, hence $K=\frac{(T-5)^{2}}{20}$. The condition that $19, K, 4 T+11$ is an increasing arithmetic sequence implies that $K-19=4 T+11-K$, hence $K=2 T+15$. With $T=55$, each of these equations implies that $K=\mathbf{1 2 5}$. Note that the two equations can be combined and solved without being passed a value of $T$. A quadratic equation results, and its roots are $T=55$ or $T=-5$. However, with $T=-5$, neither of the given sequences is increasing.

Final answer: 125