In the number series 1, 2, 8, 33, ?, find the next term that correctly continues the progression.

Introduction / Context:
This series question involves a combination of multiplication and addition, where both operations change in a regular way. The pattern is slightly more complex than simple arithmetic or geometric progressions, making it a good test of pattern recognition and algebraic thinking in aptitude exams.

Given Data / Assumptions:

• The series is: 1, 2, 8, 33, ?
• We need to find the single correct next term from the options.
• The rule is expected to be consistent and reasonably simple once identified.

Concept / Approach:
We look for a pattern that expresses each term from the previous one. A helpful idea is to see whether the multiplier and the added number are changing in a predictable way. By comparing how 1 becomes 2, 2 becomes 8 and 8 becomes 33, we can attempt to map a general rule that depends on the step number.

Step-by-Step Solution:
From 1 to 2: 1 * 1 + 1^2 = 1 + 1 = 2.From 2 to 8: 2 * 2 + 2^2 = 4 + 4 = 8.From 8 to 33: 8 * 3 + 3^2 = 24 + 9 = 33.We can see that at each step n, the operation is: next term = current term * n + n^2.For the next step, n = 4, so we use: next term = 33 * 4 + 4^2.Compute this: 33 * 4 = 132 and 4^2 = 16, so the next term = 132 + 16 = 148.

Verification / Alternative check:
To verify, we can generalize the formula: if T1 = 1, then T2 = T1 * 1 + 1^2, T3 = T2 * 2 + 2^2, T4 = T3 * 3 + 3^2 and so on. Plugging in the values sequentially reproduces the entire series: 1, 2, 8, 33 and then 148. No other option fits this rule without forcing a different and less natural formula.

Why Other Options Are Wrong:
145, 143 and 141 are close to 148 but none matches the calculated value from the identified formula. Choosing any of them would break the consistent pattern of multiplying by n and adding n^2 at each step. 152 similarly does not satisfy this rule and therefore cannot be the correct continuation of the series.

Common Pitfalls:
Many candidates look only at differences (1, 6, 25) which do not form a simple arithmetic progression. Others may try constant multipliers, which also do not work. The key is to consider that both the multiplier and the added quantity can change based on the position in the sequence. Once that is tried, the pattern becomes clear and manageable.

Final Answer:
The next number in the series is 148.