Find the wrong number in the given series : 0, 3, 9, 12, 36, 39, 42, 120, 360

Question

Accepted Answer

The correct option is 3:
The given number series is 0, 3, 9, 12, 36, 39, 42, 120, 360. Our task is to identify the number that breaks the logical progression or pattern established by the other numbers in the sequence. These types of questions require us to determine the rule that connects consecutive terms, which often involves arithmetic operations like addition, subtraction, multiplication, or division, sometimes in an alternating or increasing sequence.

Let us analyze the relationship between successive terms in the series:1.  From the first term to the second: $0 + 3 = 3$. The operation is addition of 3.
2.  From the second term to the third: $3 \times 3 = 9$. The operation is multiplication by 3.
3.  From the third term to the fourth: $9 + 3 = 12$. The operation is addition of 3.
4.  From the fourth term to the fifth: $12 \times 3 = 36$. The operation is multiplication by 3.
5.  From the fifth term to the sixth: $36 + 3 = 39$. The operation is addition of 3.The consistent pattern observed is an alternating set of operations: (+3), followed by ($	imes 3$), and this cycle repeats.

Following this established rule, the next operations should be:  * The transition from the sixth term (39) to the seventh term: The previous operation was addition (+3), so the current operation should be multiplication ($\times 3$). The expected seventh term is $39 \times 3 = 117$.
  * However, the given seventh term in the series is 42. Since $39 \times 3 \ne 42$, the number 42 is the incorrect number in the sequence.
  * To verify the rest of the series, we replace the incorrect number (42) with the correct calculated number (117). The next step in the established pattern is addition of 3. The transition from the new seventh term (117) to the eighth term: $117 + 3 = 120$. This matches the eighth term in the series.
  * The final step is multiplication by 3. The transition from the eighth term (120) to the ninth term: $120 \times 3 = 360$. This matches the final term in the series.Since the entire series fits the pattern $(+3, \times 3)$ only if the term 42 is replaced by 117, **42 (Option 3)** is the wrong number in the sequence. The series is fundamentally built on the concept of repeating, simple arithmetic progressions, and the presence of 42 violates this core rule.

The incorrect option is 1:
The number 12 is the result of the third step in the sequence ($9 + 3 = 12$) and conforms perfectly to the alternating pattern of $(\times 3)$ followed by $(+3)$. If 12 were the wrong number, the rest of the sequence from 36 onwards would not follow the established rule when working backward or forward. Therefore, 12 is a correct term in the series.

The incorrect option is 2:
The number 39 is the sixth term in the series and is derived correctly by adding 3 to the fifth term ($36 + 3 = 39$). It accurately follows the pattern following the multiplication by 3 operation that produced 36. Since 39 adheres to the $(\times 3, +3)$ alternating structure, it cannot be the wrong number.

The incorrect option is 4:
The number 360 is the final term in the sequence and is correctly generated by multiplying the eighth term (120) by 3 ($120 \times 3 = 360$). If 42 is correctly replaced by 117, the generation of 360 confirms the consistency of the pattern up to the end of the series. Therefore, 360 is a necessary and correct element of the sequence.