Final answer:
King Duncan stays at Macbeth's castle to express gratitude for Macbeth's heroism, unaware of the imminent betrayal.
Step-by-step explanation:
In William Shakespeare's play Macbeth, King Duncan plans to stay overnight at Macbeth's castle as a gesture of appreciation for Macbeth's valor and contributions to the kingdom. This is evident from the warm welcome Duncan extends towards Lady Macbeth and the praise he bestows on Macbeth for his brave deeds in battle. Duncan's visit is meant to honor Macbeth, and during his stay, he is completely unaware of the deadly intentions that await him.