Find the next number in the series 3, 10.5, 36.75, 128.625, ? by recognising the repeated multiplication pattern.

Introduction / Context:
This number series includes decimal values: 3, 10.5, 36.75, 128.625, ?. The values grow quickly, and each new term is roughly three times the previous one, but not exactly. This suggests a geometric type progression where each term is multiplied by a constant factor greater than 3.

Given Data / Assumptions:
The known terms are:

• 3
• 10.5
• 36.75
• 128.625
• ?

We assume that the same multiplicative factor is used at every step to obtain the next term, with no additional additive adjustments.

Concept / Approach:
To find the common ratio, we divide each term by the previous one. If the ratio is the same for all consecutive pairs, the series is geometric with that ratio. Once the common ratio is identified, we multiply the last known term by this ratio to obtain the next term in the series.

Step-by-Step Solution:
Step 1: Compute the ratio between the first two terms: 10.5 / 3 = 3.5.Step 2: Compute the ratio between the second and third terms: 36.75 / 10.5 = 3.5.Step 3: Compute the ratio between the third and fourth terms: 128.625 / 36.75 = 3.5.Step 4: The ratio is consistently 3.5, so the series is geometric with common ratio r = 3.5.Step 5: To find the next term, multiply the last term by 3.5: 128.625 * 3.5.Step 6: Compute 128.625 * 3 = 385.875, and 128.625 * 0.5 = 64.3125. Adding gives 385.875 + 64.3125 = 450.1875.Step 7: The given options approximate this value as 450.187, which is the closest match.

Verification / Alternative check:
Once we know the series is geometric with ratio 3.5, we can also check that 3 * 3.5 = 10.5, 10.5 * 3.5 = 36.75, and 36.75 * 3.5 = 128.625. The consistency across all these steps confirms that using the same ratio again to get the next term is valid. Therefore, 128.625 * 3.5 ≈ 450.187 is the only reasonable next value.

Why Other Options Are Wrong:
The other options 295.651, 354.146 and 398.78 do not equal 128.625 multiplied by 3.5. Using any of them as the next term would destroy the constant ratio and produce an inconsistent set of multipliers, so they are incompatible with the identified geometric pattern.

Common Pitfalls:
Some candidates may attempt to compute differences between terms and hope for a simple arithmetic pattern, but the differences here (7.5, 26.25, 91.875) are clearly not constant or linearly related. Recognising that ratios, not differences, are stable is the core skill when dealing with rapidly growing decimal sequences like this one.

Final Answer:
The next number in the geometric series is approximately 450.187.