NIMCET 2007 — Reasoning PYQ

1. Analyzing the Given Sequence

The given numerical sequence is:

$$3,51,51055,255255$$

In this type of reasoning puzzle, the numbers grow at a highly rapid rate, which strongly indicates a geometric progression or a multiplication-based sequence involving prime numbers or consecutive odd numbers. Let us find the underlying rule connecting the terms.

2. Identifying the Step-by-Step Pattern

• First Term to Second Term ($3\to 51$):

  To see what multiplier transforms $3$ into $51$, we divide:

  $$\frac{51}{3}=17$$

  This gives us the first operational rule:

  $$3\times 17=51$$

• Second Term to Third Term ($51\to \text{Expected Term}$):

  The number $17$ is a well-known prime number. Let us test if the sequence progresses by multiplying consecutive prime numbers in ascending order.

  The consecutive prime numbers starting from $17$ are:

  $$17,19,23,29,\dots$$

  Let's multiply the second term ($51$) by the next prime number, which is $19$:

  $$\text{Expected Third Term}=51\times 19=969$$

  However, $969$ does not match the given third number ($51055$), nor does it match the option choices well. Let us look at another sequence of multipliers.

• Alternative Pattern: Product of Consecutive Odd Prime Numbers

  Notice the digits of the third and fourth numbers closely. They resemble products of fundamental prime numbers:

  Let's try multiplying prime numbers directly to check if they form composite combinations:

  $$3\times 17=51$$

  The prime factors of $51$ are $3$ and $17$.

  Let's look at the product of consecutive odd prime numbers: $3\times 5\times 7\times 11\times 13\times 17\times \dots$

  Let's compute the product of consecutive odd prime numbers up to $17$:

  $$3\times 5\times 7=105$$

  $$105\times 11=1155$$

  $$1155\times 13=15015$$

  $$15015\times 17=255255$$

  Notice that $255255$ is exactly the fourth term in our sequence!

  Therefore, the sequence terms are built by progressively expanding the product of consecutive odd prime numbers:

  • $1^{\text{st}}$ Term: Given as the initial base prime number:

    $$\text{Term 1}=3$$

  • $2^{\text{nd}}$ Term: Product of the first few odd primes up to $17$ (skipping intermediate blocks) or a combined step:

    $$\text{Term 2}=3\times 17=51$$

  • $4^{\text{th}}$ Term: The product of all consecutive odd primes from $3$ to $17$:

    $$\text{Term 4}=3\times 5\times 7\times 11\times 13\times 17=255255$$

3. Spotting the Incorrect Term

Let's see how the $3^{\text{rd}}$ term should fit into this sequence chain perfectly.

If the fourth term is $255255$, let's divide it backward by the last prime factor ($17$) to find what the correct preceding third term should be:

$$\text{Correct Third Term}=\frac{255255}{17}=15015$$

Let us verify if $15015$ makes structural sense with the second term ($51$):

$$\frac{15015}{51}=295⟹5\times 59\text{ (not a single consecutive prime cluster)}$$

Let's look at the option choices provided: (a) $51$, (b) $55$, (c) $51055$, (d) $255255$.

Notice that option (b) lists $55$, but the number printed in the original series is $51055$.

This reveals a typographical pattern inside the wrong term itself! The number $51055$ was written incorrectly in place of the mathematical combination. Let's inspect the division of the given incorrect term:

$$\frac{255255}{5}=51051$$

If the sequence was generated using standard factor sub-blocks:

$$3$$

$$3\times 17=51$$

$$\text{Correct Third Term}=51\times 1001=51051(\text{Since }7\times 11\times 13=1001)$$

$$51051\times 5=255255$$

Thus, the precise correct sequence value must be $51051$. The question provides $51055$ instead, making it the mathematically flawed, incorrect term of the series.

Final Answer

The incorrect term in the series is $51055$.

Therefore, the correct option is (c).