Final answer:
Ban Gu wrote the 'Han-shu' in the 1st century BC, a chronicle that documents the Western Han dynasty and serves as a continuation of Sima Qian's historical work, the 'Shi-ji' or 'Records of the Grand Historian'.
Step-by-step explanation:
In the 1st century BC, the Chinese chronicle written by Ban Gu is known as the Han-shu. This historical work is a vital source of information regarding the Western Han dynasty. Ban Gu's extensive historical account complements the earlier work of Sima Qian, who is regarded as the father of Chinese history. While Sima Qian authored the Shi-ji or Records of the Grand Historian, it was Ban Gu who continued the tradition of historical compilation in the Han era, providing insights into the government structures, biographies, and the territorial expansion of the Han dynasty.
Option (1) Han-shu is, therefore, the correct answer to the question regarding which chronicle was written by Ban Gu. It is important to differentiate between the works of Sima Qian and Ban Gu, as the former is known for the Shi-ji and the latter for the Han-shu, both of which are paramount for our understanding of early Chinese history and historiography.