Step 1: Concept Identification: The problem involves a mixed series where each term consists of a letter, a number, and another letter. The objective is to determine the outlier by analyzing the pattern of each component individually.
Step 2: Analytical Framework:
1. Examine the sequence of the first letter in each term.
2. Examine the sequence of the last letter in each term.
3. Examine the sequence of the numbers in each term.
Step 3: Detailed Analysis:
The given series is: H4Q, K10N, N20K, Q43H, T90E.
First Letter Pattern:
H (8) $\xrightarrow{+3}$ K (11) $\xrightarrow{+3}$ N (14) $\xrightarrow{+3}$ Q (17) $\xrightarrow{+3}$ T (20).
This progression is consistent.
Last Letter Pattern:
Q (17) $\xrightarrow{-3}$ N (14) $\xrightarrow{-3}$ K (11) $\xrightarrow{-3}$ H (8) $\xrightarrow{-3}$ E (5).
This progression is also consistent.
Number Pattern:
The sequence of numbers is: 4, 10, 20, 43, 90.
A recursive relationship of the form \(a_n = k \cdot a_{n-1} + c\) is suspected. Testing with k=2.
Assuming a rule \(a_n = 2 \times a_{n-1} + (n-1)\) and starting from the first term:
Term 1 number = 4.
Expected Term 2 number: \(2 \times 4 + (2-1) = 8 + 1 = 9\). The series shows 10, indicating K10N may be the outlier.
Verifying the pattern with the assumed correct second number (9):
Expected Term 3 number: \(2 \times 9 + (3-1) = 18 + 2 = 20\). Matches N20K.
Expected Term 4 number: \(2 \times 20 + (4-1) = 40 + 3 = 43\). Matches Q43H.
Expected Term 5 number: \(2 \times 43 + (5-1) = 86 + 4 = 90\). Matches T90E.
The pattern \(a_n = 2a_{n-1} + n-1\) holds for all terms except the second.
Step 4: Conclusion:
The term K10N is the outlier because its numerical component should be 9, not 10, to adhere to the established pattern.