A number series is given with one term missing. Choose the correct alternative from the given options that will complete the pattern and identify the missing number in the sequence: 5, 10, 13, 52, 57, ?

Introduction / Context:
This question presents a mixed operation number series. The terms 5, 10, 13, 52, 57, ? alternate between moderate jumps and very large jumps, hinting that the pattern uses both multiplication and addition in a repeating cycle. Such alternating operation series are common in reasoning tests because they require careful observation over several steps.

Given Data / Assumptions:

• Series: 5, 10, 13, 52, 57, ?
• Options: 285, 225, 342, 390.
• We assume that the same alternating pattern continues throughout the series.

Concept / Approach:
When terms increase sharply at every second step, a common pattern is "multiply by a certain number, then add another number" in an alternating way. Here we check transitions in order and see if the multipliers and addends form a simple arithmetic or geometric pattern, such as 2, 4, 6 or 3, 5, 7 and so on. Once found, we extend that pattern to compute the missing term.

Step-by-Step Solution:
Step 1: Examine transitions closely.From 5 to 10: 5 * 2 = 10.From 10 to 13: 10 + 3 = 13.From 13 to 52: 13 * 4 = 52.From 52 to 57: 52 + 5 = 57.Step 2: Recognise the pattern.Operations alternate as multiply, add, multiply, add and so on.Multipliers: 2, then 4, then next should be 6.Addends: 3, then 5, which are consecutive odd numbers, so the next addend would be 7 in future steps.Step 3: Apply the next operation to find the missing term.After 57, the next operation in the established pattern is multiply by 6.Missing term = 57 * 6 = 342.

Verification / Alternative check:
Write the series together with the operations: 5 * 2 = 10, 10 + 3 = 13, 13 * 4 = 52, 52 + 5 = 57, 57 * 6 = 342. We can also predict the next step for confirmation: 342 + 7 = 349 if the series continued. The pattern is therefore multiply by an even number (2, 4, 6, ...) and then add the next odd number (3, 5, 7, ...), alternating these operations. This confirms that 342 is consistent.

Why Other Options Are Wrong:
285: Not equal to 57 multiplied by any simple continuation of the multiplier pattern, and does not fit an add operation either.225: Similarly fails the multiply by 6 or add 7 expectation and breaks the structure.390: Would correspond to 57 * something that does not continue the simple even number sequence 2, 4, 6.

Common Pitfalls:
Many candidates try to find one single operation linking every step, such as adding increasing differences only. However, because the series alternates operations, focusing on just differences can become confusing. A better method is to inspect each pair of terms, hypothesise a candidate operation, and then check whether that operation alternates regularly through the series.

Final Answer:
The missing number, obtained by multiplying 57 by 6, is 342.