What number should replace the question mark in the series 38, 195, ?, 4905, 24530, 122655?

Introduction / Context:
This number series question involves rapidly increasing values that suggest a multiplicative pattern. The challenge is to detect the specific operation used at each step and then apply it to find the missing term.

Given Data / Assumptions:

• Partial series: 38, 195, ?, 4905, 24530, 122655.
• Exactly one term in the middle is missing.
• The series appears to grow roughly by a factor of about five each time.

Concept / Approach:
Since the numbers grow quite quickly, start by checking whether each term relates to the previous term via a constant multiplier or a simple linear expression such as multiplying by a fixed number and then adding or subtracting a small constant.

Step-by-Step Solution:
Step 1: Examine the relation between the first two terms: 38 to 195. Step 2: Compute 38 * 5 = 190, and 190 + 5 = 195. So 195 = 38 * 5 + 5. Step 3: Assume the same rule applies: term_{n+1} = term_n * 5 + 5. Step 4: Let the missing term be X. Then X should be 195 * 5 + 5. Step 5: Calculate 195 * 5 = 975 and 975 + 5 = 980, so X = 980. Step 6: Check from X to the next known term: 980 * 5 + 5 = 4900 + 5 = 4905 which matches the series. Step 7: Continue for completeness: 4905 * 5 + 5 = 24525 + 5 = 24530; 24530 * 5 + 5 = 122650 + 5 = 122655. The rule holds throughout.

Verification / Alternative check:
By rebuilding the series with the rule term_{n+1} = term_n * 5 + 5, we get the full list: 38, 195, 980, 4905, 24530, 122655. Every step is consistent, so there is no contradiction. This confirms that 980 is the only value that keeps the pattern intact.

Why Other Options Are Wrong:
If we choose 885, 824, or 1021, the relation term_{n+1} = term_n * 5 + 5 fails at one or more positions. The products and sums no longer match the given terms 4905, 24530, or 122655, so those options break the series rule.

Common Pitfalls:
A typical mistake is to approximate the factor as 4 or 6 because of mental rounding, which leads to inconsistent patterns and incorrect answers. Another pitfall is to forget the constant addition after multiplication and check only for pure multiples of 5. Always verify the pattern across multiple adjacent terms.

Final Answer:
The missing number that correctly completes the series is 980.