Find the missing term in the series 2, 3, 6, 15, ?, 157.5.

Introduction / Context:
This problem involves a multiplicative number series where the multiplying factor itself changes according to a simple pattern. Recognising this pattern is the key to identifying the missing term.

Given Data / Assumptions:

• Series: 2, 3, 6, 15, ?, 157.5.
• Exactly one term in the middle is missing.
• Decimal result at the end suggests fractional multipliers.

Concept / Approach:
To solve this, we examine the ratio between consecutive terms to see whether there is a sequence of multipliers with a regular increment or decrement. Since the numbers grow quite quickly but not as fast as a pure geometric progression with a fixed ratio, we suspect that the multiplier itself is changing gradually.

Step-by-Step Solution:
Step 1: Find the factor from 2 to 3: 3 / 2 = 1.5. Step 2: From 3 to 6 the factor is 6 / 3 = 2. Step 3: From 6 to 15 the factor is 15 / 6 = 2.5. Step 4: The multipliers so far are 1.5, 2.0, 2.5, suggesting that each time the factor increases by 0.5. Step 5: The next multiplier should therefore be 3.0. Step 6: Apply this to the known term 15: 15 * 3 = 45, so the missing term is 45. Step 7: Check the factor from 45 to 157.5: 157.5 / 45 = 3.5, which continues the pattern of multipliers 1.5, 2.0, 2.5, 3.0, 3.5.

Verification / Alternative check:
Rewriting the entire series with the discovered rule yields: start with 2, then multiply successively by 1.5, 2, 2.5, 3, and 3.5. This results in 2, 3, 6, 15, 45, 157.5. Every step fits the established sequence of increasing multipliers, which confirms the answer.

Why Other Options Are Wrong:
Values like 87, 78.5, or 56.5 break the smooth pattern of multipliers increasing by 0.5. For example, if the missing term were 87, then 87 / 15 is 5.8 which does not fit the half step pattern, and 157.5 / 87 is also inconsistent. Therefore those values cannot maintain the intended rule.

Common Pitfalls:
Many learners look only for integer multipliers or fixed ratios and may give up when they see decimals. Others try to force an additive pattern even when the numbers grow too rapidly for that to make sense. Always be ready to consider fractional multipliers and check whether the changes in the multiplier follow a simple progression.

Final Answer:
The missing term that continues the increasing half step multiplier pattern is 45.