Let $T \ge 1$ be some finite integer. Solve the following maximization problem: $$\text{Maximize} \sum_{t=1}^T \left(\frac12 \right)^t \sqrt{x_t}$$ $$\text{subject to} \sum_{t=1}^T x_t \le 1$$ $$\text{and} \quad x_t\ge0, t=1, ... , T.$$
I was trying to use the Kuhn-Tucker theorem to prove this. First, $L(x_1, ... , x_t, \lambda_1, ... , \lambda_{T+1}) = \sum_{i=1}^T \left( \frac12 \right)^t \sqrt{x_t} + \lambda_1(1- \sum_{t=1}^T x_t) + \sum_{t=1}^T \lambda_{t+1}x_t$. I need to find the critical point of this function. $\frac{\partial L}{\partial x_t } = \left(\frac12 \right)^{t+1} \frac1{\sqrt{x_t}} - \lambda_1 + \lambda_{t+1} = 0 $ for $t=1 , ... , T$, and $\frac{\partial L}{\partial \lambda_1} = 1- \sum_{i=1}^T x_t$, and $\frac{\partial L}{\partial\lambda_t} = x_{t-1} $ for $t=2, ... , T+1$. I am only familiar with the constrained optimization problem with $2$ or $3$ constraints. In this case, I can consider each case that one constraint is effective and another is not and so on. But since there are too many constraints here, I do not know how to deal with this. I appreciate if you give some help.
HINT: Since for all $t$ you have $x_t \ge 0$, you can call $$y_t= \sqrt {x_t}$$
So that your problem becomes: maximize the quantity $$\sum_{t=1}^T 2^{-t}y_t$$ under the constraint $$y_1^2+ \dots + y_T^2 \le 1$$ This is much more manageable, since those ugly square roots disappear.