In the following number series 2, 3, 6, 15, 45, what number should replace the question mark as the next term in the sequence?

Introduction / Context:
Number series questions are common in aptitude tests and are used to check pattern recognition skills. The given series 2, 3, 6, 15, 45, ? is a classic example where the pattern involves changing multiplication factors. The goal is to identify how each term is generated from the previous term and then extend the pattern to find the missing value.

Given Data / Assumptions:
- The series is: 2, 3, 6, 15, 45, ?. - Each term is obtained from the previous term using a consistent rule. - We assume there is a single underlying pattern, not a random jump.

Concept / Approach:
- First, look at the ratios or multipliers between consecutive terms. - Compute 3 / 2, 6 / 3, 15 / 6 and 45 / 15 to detect any trend in the multiplying factors. - If the multipliers follow a systematic pattern, extend that pattern to predict the next multiplier and hence the next term.

Step-by-Step Solution:
Step 1: Find the multiplier from 2 to 3. Here 2 * 1.5 = 3. Step 2: From 3 to 6, the multiplier is 2, since 3 * 2 = 6. Step 3: From 6 to 15, the multiplier is 2.5, since 6 * 2.5 = 15. Step 4: From 15 to 45, the multiplier is 3, since 15 * 3 = 45. Step 5: The sequence of multipliers is 1.5, 2, 2.5, 3. This clearly increases by 0.5 each time. Step 6: Following this pattern, the next multiplier should be 3.5. Step 7: Therefore, the next term is 45 * 3.5 = 45 * (7 / 2) = (45 * 7) / 2 = 315 / 2 = 157.5.

Verification / Alternative check:
We can verify by writing the general idea: each term is obtained from the previous one by multiplying with a factor that increases by 0.5 successively. So term n has multiplier 1.5 + 0.5 * (n - 2) when n starts from 2. Checking all current steps confirms the pattern, so using 3.5 for the next step is fully justified.

Why Other Options Are Wrong:
Option A (90): would require a multiplier of 2 from 45 to 90, which breaks the 0.5 increment pattern. Option B (135): would require a multiplier of 3 from 45 to 135, but we have already used 3 from 15 to 45 and the pattern demands 3.5 next. Option D (200): implies a multiplier of 4.44 which does not follow any simple regular increase. Option E (210): would correspond to a multiplier of 4.666 which again does not fit the simple arithmetic progression of multipliers.

Common Pitfalls:
- Many learners look only at differences of terms rather than ratios, which does not reveal the pattern in this case. - Some may assume a fixed multiplier, which fails once they see 2 to 3 and 3 to 6 are different. - Another mistake is to stop after noticing an approximate pattern rather than checking all transitions to confirm consistency.

Final Answer:
The next term in the series that follows the multiplier pattern is 157.5.