In the following, all terms of the sequences under consideration are real numbers.

(1) For a geometric sequence $\left\{ s _ { n } \right\}$ with first term 1 and common ratio 2,

$$s _ { 1 } s _ { 2 } s _ { 3 } = \square , \quad s _ { 1 } + s _ { 2 } + s _ { 3 } = \square$$

(2) Let $\left\{ s _ { n } \right\}$ be a geometric sequence with first term $x$ and common ratio $r$. Let $a, b$ be real numbers (with $a \neq 0$), and suppose the first three terms of $\left\{ s _ { n } \right\}$ satisfy

$$\begin{aligned}
& s _ { 1 } s _ { 2 } s _ { 3 } = a ^ { 3 } \\
& s _ { 1 } + s _ { 2 } + s _ { 3 } = b
\end{aligned}$$

Then

$$x r = \square$$

Furthermore, using (2) and (3), we find the relation satisfied by $r, a, b$:

$$\text { エ } r ^ { 2 } + ( \text { オ } - \text { カ } ) r + \text { 倍 } = 0$$

Since there exists a real number $r$ satisfying (4),

$$\text { ク } a ^ { 2 } + \text { ケ } a b - b ^ { 2 } \leqq 0$$

Conversely, when $a, b$ satisfy (5), we can find the values of $r, x$ using (3) and (4).

(3) When $a = 64 , b = 336$, consider the geometric sequence $\left\{ s _ { n } \right\}$ satisfying conditions (1) and (2) in (2) with common ratio greater than 1. Using (3) and (4), we find the common ratio $r$ and first term $x$ of $\left\{ s _ { n } \right\}$: $r = \square , x =$ サシ.

Using $\left\{ s _ { n } \right\}$, define the sequence $\left\{ t _ { n } \right\}$ by

$$t _ { n } = s _ { n } \log _ { \square } s _ { n } \quad ( n = 1,2,3 , \cdots )$$

Then the general term of $\left\{ t _ { n } \right\}$ is $t _ { n } = ( n +$ ス $) \cdot$ コ $^ { n + }$ セ. The sum $U _ { n }$ of the first $n$ terms of $\left\{ t _ { n } \right\}$ is found by computing $U _ { n } - \square U _ { n }$: