Explanation:

To find the missing letter cluster, we need to analyze the pattern of transformation for each letter position across the series. \nStep 1: Analyze the first letter of each cluster.\nE ($5^{th}$) $\rightarrow$ G ($7^{th}$) $\rightarrow$ I ($9^{th}$) $\rightarrow$ ? $\rightarrow$ M ($13^{th}$)\nThe pattern is an increase of $2$ for each step: $5+2=7$, $7+2=9$.\nFollowing this pattern, the next first letter will be I ($9$) $+ 2 = 11^{th}$ letter, which is K.\nTo verify, K ($11$) $+ 2 = 13^{th}$ letter, which is M. This confirms the pattern for the first letter. \nStep 2: Analyze the second letter of each cluster.\nQ ($17^{th}$) $\rightarrow$ R ($18^{th}$) $\rightarrow$ S ($19^{th}$) $\rightarrow$ ? $\rightarrow$ U ($21^{st}$)\nThe pattern is an increase of $1$ for each step: $17+1=18$, $18+1=19$.\nFollowing this pattern, the next second letter will be S ($19$) $+ 1 = 20^{th}$ letter, which is T.\nTo verify, T ($20$) $+ 1 = 21^{st}$ letter, which is U. This confirms the pattern for the second letter. \nStep 3: Analyze the third letter of each cluster.\nM ($13^{th}$) $\rightarrow$ L ($12^{th}$) $\rightarrow$ K ($11^{th}$) $\rightarrow$ ? $\rightarrow$ I ($9^{th}$)\nThe pattern is a decrease of $1$ for each step: $13-1=12$, $12-1=11$.\nFollowing this pattern, the next third letter will be K ($11$) $- 1 = 10^{th}$ letter, which is J.\nTo verify, J ($10$) $- 1 = 9^{th}$ letter, which is I. This confirms the pattern for the third letter. \nStep 4: Combine the determined letters.\nThe missing letter cluster is K T J. \nStep 5: Evaluate the options.\nOption A: JTK. Incorrect.\nOption B: KTJ. This matches our derived cluster.\nOption C: JVK. Incorrect.\nOption D: KVJ. Incorrect. \nTherefore, the correct option is 'KTJ'.