Choose the correct alternative that continues the pattern and replaces x in the series: 5, 15, 17, 85, 89, 623, x.

Introduction / Context:
This series alternates between multiplication and addition. The multipliers and addends themselves follow a simple pattern involving consecutive odd and even numbers. Recognising such alternating operation patterns is a key skill in quantitative aptitude sections.

Given Data / Assumptions:

• Given series: 5, 15, 17, 85, 89, 623, x.
• Only the last term is missing.
• Operations between terms seem to alternate between large and small jumps.

Concept / Approach:
We test a hypothesis that the series alternates between multiplying by an odd integer and then adding an even integer, or vice versa. The best way is to compute the change from one term to the next and see if the pattern in operations repeats consistently across the series.

Step-by-Step Solution:
Step 1: Compare 5 and 15. We can write 5 × 3 = 15. Step 2: Compare 15 and 17. We have 15 + 2 = 17. Step 3: Compare 17 and 85. We get 17 × 5 = 85. Step 4: Compare 85 and 89. We have 85 + 4 = 89. Step 5: Compare 89 and 623. We see 89 × 7 = 623. Step 6: The pattern is clear: multiply by 3, add 2, multiply by 5, add 4, multiply by 7, and the next step should be add 6. Step 7: Calculate the missing term after 623 by adding 6: 623 + 6 = 629.

Verification / Alternative check:
Write the complete sequence with operations: 5 × 3 = 15, 15 + 2 = 17, 17 × 5 = 85, 85 + 4 = 89, 89 × 7 = 623, 623 + 6 = 629. The multipliers are odd numbers increasing by 2 (3, 5, 7), and the addends are even numbers increasing by 2 (2, 4, 6). This perfectly structured alternation confirms that 629 is the correct next term.

Why Other Options Are Wrong:
Values such as 825, 759, and 675 cannot be obtained from 623 through the operation of adding 6. If we try to force them into the pattern, we would have to change either the odd multipliers or the even addends, destroying the neat structure. Because exam questions aim for simple, predictable rules, any option that does not follow this alternating scheme must be rejected.

Common Pitfalls:
A typical error is to look only at first differences, which here are 10, 2, 68, 4, 534 and appear chaotic. Another mistake is to assume a single type of operation (all multiplication or all addition) instead of considering an alternating mix. Always examine at least three consecutive steps to see if two different operations are recurring in a pattern.

Final Answer:
Applying the rule of alternately multiplying by odd numbers and adding increasing even numbers, the value of x is 629.