In the series 2.5, 3.5, 9, 40, ?, which number should come next to follow the underlying multiplication-and-addition pattern?

Introduction / Context:
This number series problem involves non-integer terms and a pattern built from multiplication by powers of 2 followed by a small addition. Recognizing that both the multiplier and the added value change systematically is the key to identifying the next term in the series.

Given Data / Assumptions:
The series is: 2.5, 3.5, 9, 40, ?. You must choose the correct next term from the options 316, 328, 323 and 340. We assume the series uses a consistent rule where each term is obtained by multiplying the previous term by a certain integer and then adding that same integer.

Concept / Approach:
A good approach is to check ratios and differences. The jump from 2.5 to 3.5 is small, but from 3.5 to 9 and then to 40 it is much larger, suggesting multiplication by increasing integers. We test whether the pattern “multiply by a power of 2 and then add that same power of 2” holds: multiply by 1 and add 1, then multiply by 2 and add 2, then multiply by 4 and add 4, and so on.

Step-by-Step Solution:
Step 1: From 2.5 to 3.5. We can write this as 2.5 * 1 + 1 = 2.5 + 1 = 3.5.Step 2: From 3.5 to 9. Check 3.5 * 2 + 2 = 7 + 2 = 9. This matches, with multiplier 2 and addition 2.Step 3: From 9 to 40. Check 9 * 4 + 4 = 36 + 4 = 40. This matches, with multiplier 4 and addition 4.Step 4: Observe the pattern in the multipliers and addends: 1, 2, 4. These are powers of 2 (2^0, 2^1, 2^2). The next power of 2 is 8 (2^3).Step 5: Therefore, the next term should be 40 * 8 + 8 = 320 + 8 = 328.

Verification / Alternative check:
We can summarize the rule as: next term = current term * k + k, where k takes values 1, 2, 4, 8, ... This pattern is regular and uses simple powers of 2. Substituting k = 8 for the step after 40 gives 328. None of the other options maintain this “multiply by k and add k with k doubling each time” structure.

Why Other Options Are Wrong:
316, 323 and 340 do not equal 40 * 8 + 8. They would require inconsistent multipliers and addends that do not follow the earlier pattern of 1, 2, 4. Choosing any of them would break the clean power-of-2 based rule, so they cannot be correct.

Common Pitfalls:
Candidates may only look for additive patterns (such as adding 1, then 5, then 31) and miss the multiplicative structure. Others may not recognize that the multipliers 1, 2 and 4 form a doubling sequence. Checking both multiplication and addition together is very helpful when jumps between terms grow quickly.

Final Answer:
The number that correctly continues the pattern is 328.