Find the next term in the series 3, 6, 17.5, 64.75, ?

Introduction / Context:

This question presents a non standard number series with decimal values and rapidly increasing terms. Such series are meant to test advanced pattern recognition, particularly when both the multiplicative factor and an additive constant appear to change together. Candidates must be comfortable working with fractional multipliers and decimals, and must systematically test possible rules instead of guessing by rough magnitude alone.

Given Data / Assumptions:

• The series is 3, 6, 17.5, 64.75, ?.
• We must find the next term in the sequence.
• All terms after the first are larger than their predecessors, and decimals appear from the third term onward.
• The pattern likely involves multiplication by gradually increasing factors along with a related additive component.

Concept / Approach:

Rather than looking at simple differences, which are irregular here, it is more effective to see how each term can be expressed as the previous term multiplied by a factor and then adjusted by a constant. A closely related pattern is that each step multiplies by n point five and adds the same n point five, where n increases by one each time. This structure, multiply then add by the same half integer, explains the decimal nature of the series and provides a consistent way to compute subsequent terms.

Step-by-Step Solution:

Step 1: From 3 to 6: 3 * 1.5 + 1.5 = 4.5 + 1.5 = 6, so the factor and addition are both 1.5. Step 2: From 6 to 17.5: 6 * 2.5 + 2.5 = 15 + 2.5 = 17.5, with factor and addition both 2.5. Step 3: From 17.5 to 64.75: 17.5 * 3.5 + 3.5 = 61.25 + 3.5 = 64.75, with factor and addition both 3.5. Step 4: The pattern is clear: at each step we multiply by (n + 0.5) and then add (n + 0.5), where n takes values 1, 2, 3, and so on. Step 5: For the next term, use n = 4, so factor and addition are 4.5. Compute 64.75 * 4.5 + 4.5 = 291.375 + 4.5 = 295.875.

Verification / Alternative check:

We can rewrite each term using the pattern a(n+1) = a(n) * (n + 0.5) + (n + 0.5). With a(1) = 3, we get a(2) = 3 * 1.5 + 1.5 = 6, a(3) = 6 * 2.5 + 2.5 = 17.5, a(4) = 17.5 * 3.5 + 3.5 = 64.75, a(5) = 64.75 * 4.5 + 4.5 = 295.875. Every term matches the provided series up to the fourth one, and the calculation for the fifth term smoothly extends the same rule. No other candidate value yields such a perfect continuation with this consistent structure.

Why Other Options Are Wrong:

Option 304.587 cannot be formed by multiplying 64.75 by any half integer of the observed type and then adding the same value. Option 115.246 is far too small relative to the established growth and does not match any logical factor plus the same addend. Option 101 similarly fails, as 64.75 multiplied by a factor around 1.5 or 2.5 would not align with earlier steps. Option 250.375 also breaks the clear pattern of using factors 1.5, 2.5, 3.5, 4.5 and adding the same half integer. Only 295.875 is generated exactly from 64.75 using factor and addition 4.5.

Common Pitfalls:

Many candidates first attempt to use constant or integer ratios, which quickly fail because the ratios between terms here are not integers. Others look purely at differences, but the successive differences do not follow a simple progression. Some learners hesitate when decimals appear and assume a very complicated or arbitrary rule. Recognizing that half integer multipliers, such as 1.5, 2.5, 3.5, are natural extensions of integer patterns enables a clean solution. Practicing such mixed integer and decimal series improves confidence with these question types.

Final Answer:

The next term in the series, obtained by multiplying 64.75 by 4.5 and adding 4.5, is 295.875.