Find the next number in the number series 1, 4, 12, 32, ? by identifying the pattern of doubling and adding growing powers of 2.

Introduction / Context:
This question gives a number series and asks you to determine the next term. The terms grow quite quickly: 1, 4, 12, 32, ?. Recognizing how each term is obtained from the previous one, especially using multiplication and powers of 2, is essential for solving many number series questions in aptitude tests.

Given Data / Assumptions:
The series is:

• 1
• 4
• 12
• 32
• ?

We assume a consistent rule that uses basic operations such as multiplication and addition, without any hidden or overly complex functions.

Concept / Approach:
When numbers grow quite fast but not purely by a constant multiplier, a useful approach is to check whether each term is obtained by doubling the previous term and then adding a quantity that also follows a pattern. In this series, the extra quantities added after doubling turn out to be powers of 2 that also increase systematically.

Step-by-Step Solution:
Step 1: Move from 1 to 4. Check 1 * 2 + 2 = 4. Here, we double 1 and add 2.Step 2: Move from 4 to 12. Check 4 * 2 + 4 = 12. We double 4 and add 4.Step 3: Move from 12 to 32. Check 12 * 2 + 8 = 32. We double 12 and add 8.Step 4: Observe the added numbers: 2, 4, 8. These are powers of 2, namely 2^1, 2^2 and 2^3.Step 5: The pattern suggests that the next term is found by again doubling the previous term and adding the next power of 2, which is 2^4 = 16.Step 6: Therefore, the next term should be 32 * 2 + 16 = 64 + 16 = 80.

Verification / Alternative check:
We can restate the sequence as: a(1) = 1, and for n ≥ 2, a(n) = 2 * a(n - 1) + 2^(n - 1). Applying this formula: a(2) = 2 * 1 + 2^1 = 4, a(3) = 2 * 4 + 2^2 = 12, a(4) = 2 * 12 + 2^3 = 32, and a(5) = 2 * 32 + 2^4 = 80. This confirms that 80 is the unique correct continuation.

Why Other Options Are Wrong:
100, 192 and 169 do not satisfy the consistent rule of doubling and adding successive powers of 2. For example, 100 would require adding 36 after doubling 32, and 36 does not fit into the neat sequence 2, 4, 8, 16. Therefore, those options are not logically supported by the pattern in the given terms.

Common Pitfalls:
Some students look only at differences between terms (3, 8, 20) and do not realize that these differences themselves are related to powers of 2. Others might guess from approximate growth instead of verifying a precise rule. Always try to check both multiplication and a secondary addition when growth is not purely geometric.

Final Answer:
The next number in the series, following the pattern of doubling and adding powers of 2, is 80.