109北科大電子_工數 

ㆍ(12%) Find an explicit formula for the sequences defined recursively by

(a) (4%) What are the values of G4 and G5?

(b) (5%) Give a matrix A such that

(c) (5%) Find an explicit formula for Gk.

(d) (6 %) What is the limit of Gk as k →∞?

105交大資訊聯招 (6points) Three persons, TC, CC, and YJ, have their own gold. They always divide their own two equal parts and donate to others every day. IC, for example, has 12-kilograms gold today and donate 6 kilograms to Cc and the other 6 kilograms to Y.. If TC has 6 kilograms, CC has 1 kilogram, and YJ has 2 kilograms at first, how many kilograms of gold do CC and YJ have after 365 days?

106交大資訊聯招 (10 points) Political affiliations change from one generation to the next. We consider three parties, labeled D, K, and T. In one generation, membership in D goes 80% to D, 10% to K, and 10% to T； membership in K goes 20% to D, 80% to K, and none to T； and membership in T goes 50% to D, 10% to K, and 40% to T. What is the long-term steady state for the three parties (membership ratios among these three parties)?

(a) (12%) Write down the transition matrix A and calculate its eigenvaluesλ. 

(b) (6%) Assume that the initial market share vector of A, B, and C is，what is the limit as k → ∞? 

說明：請將以下短文中劃底線處之中文句子譯成正確、通順、達意的英文(一)中譯英(8 分)① 近年來，許多發展中國家經歷了社會與經濟的巨大危機。 社會經濟困境改變了能 源方面的投資，並因移動限制嚴重影響產業發展。大多數投資活動因而面臨中斷。 ② 雖然面對政治緊張所帶來的紛亂，有些公司年營收仍創歷史新高。

(二)英譯中(8 分)        Playing sports has a major influence on a person’s health. It brings good qualities to one’s life. ③ Doing exercise properly and correctly and achieving a fitness goal can strengthen sportsmen’s core muscles and boost their athletic performance. However, sporting activities have both good and disruptive effects on one’s health. ④Positive effects from workouts are accomplished primarily through physical activities, but negative impacts of excessive movements include increased injuries and extreme fatigue.