Analysing the Letter Pattern

The series consists of letter-number clusters. First, let's analyze the pattern in the letters:

• First Letters: H, L, P, T, X. The positions in the alphabet are 8, 12, 16, 20, 24. The difference between consecutive positions is +4 (12-8=4, 16-12=4, etc.). The next first letter will be at position 24 + 4 = 28. Since the alphabet has 26 letters, position 28 corresponds to the 2nd letter, which is B.
• Second Letters: Y, C, G, K, O. The positions are 25, 3, 7, 11, 15. The difference is +4, wrapping around the alphabet (25+4=29, which is 3; 3+4=7, etc.). The next second letter will be at position 15 + 4 = 19. The 19th letter is S.
• Third Letters: P, T, X, B, F. The positions are 16, 20, 24, 2, 6. The difference is +4, wrapping around the alphabet (24+4=28, which is 2; 2+4=6, etc.). The next third letter will be at position 6 + 4 = 10. The 10th letter is J.

Combining these, the next letter cluster is BSJ.

Analysing the Number Pattern

Now, let's look at the sequence of numbers: 8, 15, 29, 57, 113.

We can observe a pattern where each number is double the previous number minus 1:

• $8 \times 2 - 1 = 16 - 1 = 15$
• $15 \times 2 - 1 = 30 - 1 = 29$
• $29 \times 2 - 1 = 58 - 1 = 57$
• $57 \times 2 - 1 = 114 - 1 = 113$

To find the next number, we apply the same pattern:

• $113 \times 2 - 1 = 226 - 1 = 225$

Alternatively, consider the differences between consecutive numbers:

• $15 - 8 = 7$
• $29 - 15 = 14$ ($7 \times 2$)
• $57 - 29 = 28$ ($14 \times 2$)
• $113 - 57 = 56$ ($28 \times 2$)

The next difference should be $56 \times 2 = 112$.

Therefore, the next number is $113 + 112 = 225$.

Determining the Missing Term

By combining the letter pattern (BSJ) and the number pattern (225), the complete next term in the series is BSJ 225.