NIMCET 2007 — Reasoning PYQ

1. Analyzing the Sequence Tendency

The given numerical sequence is steadily decreasing:

$$1015,799,674,615,583,575$$

To understand the progression, let us calculate the differences between each consecutive pair of numbers.

2. Computing the Step-by-Step Differences

Let's look at the subtraction gaps from right to left (or left to right) to spot a pattern based on squares, cubes, or geometric multiples:

• Difference 1 ($1015\to 799$):

  $$1015-799=216$$

• Difference 2 ($799\to 674$):

  $$799-674=125$$

• Difference 3 ($674\to 615$):

  $$674-615=59$$

• Difference 4 ($615\to 583$):

  $$615-583=32$$

• Difference 5 ($583\to 575$):

  $$583-575=8$$

3. Spotting the Mathematical Pattern

Let's analyze the calculated difference sequence:

$$216,125,59,32,8$$

Notice that the distinct values at the boundaries are perfect cubes of consecutive integers:

• $$216=6^3$$

• $$125=5^3$$

• $$8=2^3$$

This reveals that the sequence drops progressively by the cubes of consecutive decreasing numbers ($6^3,5^3,4^3,3^3,2^3$).

Let us re-verify what the middle differences should be according to this perfect cube rule:

• Expected third difference should be $4^3=64$ (instead of $59$).

• Expected fourth difference should be $3^3=27$ (instead of $32$).

4. Finding the Flawed Term

Let's apply the correct mathematical cube differences down the chain starting from the verified second term ($799$):

1. Start: $1015$

2. Next: $1015-6^3=1015-216=799$ (Correct)

3. Next: $799-5^3=799-125=674$ (Correct)

4. Next:

   $$674-4^3=674-64=610$$

5. Next:

   $$610-3^3=610-27=583$$

   (Matches the given term)

6. Next:

   $$583-2^3=583-8=575$$

   (Matches the given term)

The given sequence features $615$ instead of the mathematically accurate value $610$. This single error accounts for both incorrect intermediate differences ($59$ and $32$).

Final Answer

The incorrect term in the series is $615$.

Therefore, the correct option is (b).