Find the next number in the sequence: 1, 2, 6, 21, 88, ?

Introduction / Context:
This is a classic number series where each term is generated from the previous one using both multiplication and addition. Recognising when the multiplier and the added value themselves form a pattern is crucial for solving such questions quickly in examinations.

Given Data / Assumptions:

• Given sequence: 1, 2, 6, 21, 88, ?
• Only the next term is missing.
• The growth is faster than linear, suggesting multiplicative operations combined with addition.

Concept / Approach:
Look at how each term transforms into the next. When the differences alone seem irregular, consider expressions of the form previous term times some integer plus another integer, especially patterns like n and n added again. A very common construction is “multiply by n, then add n”.

Step-by-Step Solution:
Step 1: Compare 1 and 2. We can write 1 × 1 + 1 = 2. Step 2: Compare 2 and 6. We have 2 × 2 + 2 = 6. Step 3: Compare 6 and 21. We get 6 × 3 + 3 = 21. Step 4: Compare 21 and 88. We obtain 21 × 4 + 4 = 88. Step 5: The pattern is clear: at each step multiply by an increasing integer n and then add the same integer n, where n = 1, 2, 3, 4, ... Step 6: To get the next term, use n = 5: 88 × 5 + 5. Step 7: Compute 88 × 5 = 440, then 440 + 5 = 445.

Verification / Alternative check:
Write the sequence with the operations: 1 × 1 + 1 = 2, 2 × 2 + 2 = 6, 6 × 3 + 3 = 21, 21 × 4 + 4 = 88, 88 × 5 + 5 = 445. Each step uses the next natural number as both the multiplier and the addend. This perfectly explains all existing terms and produces the next one without any inconsistency.

Why Other Options Are Wrong:
Values 364, 296, and 194 cannot be expressed as 88 × 5 + 5. If we try to back fit them into a similar rule, the sequence of multipliers and addends becomes irregular or non integer, which is highly unlikely for a well designed exam question. Only 445 preserves the simple and elegant “multiply by n and add n” rule throughout the series.

Common Pitfalls:
Learners sometimes focus only on differences: 1, 4, 15, 67, which do not yield an obvious simple rule. It is more productive to study how each term arises from the previous one using both multiplication and addition. Whenever numbers grow quite quickly, suspect multiplicative steps combined with straightforward additions.

Final Answer:
Following the pattern 88 × 5 + 5, the next term in the sequence is 445.