We need to find the mathematical pattern governing the series. Let's test the formula n³ - n² (n cubed minus n squared), starting from n=2. 2³ - 2² = 8 - 4 = 4 3³ - 3² = 27 - 9 = 18 4³ - 4² = 64 - 16 = 48 5³ - 5² = 125 - 25 = 100 6³ - 6² = 216 - 36 = 180 7³ - 7² = 343 - 49 = 294 The series provided in the question lists the last number as 295. However, according to the established pattern, it should be 294. Therefore, 295 is the incorrect term.

This sequence is a "twin series" or alternating series, where two distinct patterns are interleaved. Series 1 (Odd positions: 1st, 3rd, 5th, 7th): 5, 10, 20, ? The pattern here is "multiply by 2." (5 × 2 = 10; 10 × 2 = 20). Series 2 (Even positions: 2nd, 4th, 6th): 8, 16, 32... The pattern here is also "multiply by 2." The question asks for the 7th term, which belongs to Series 1. Following the pattern of Series 1, the next number after 20 is 20 × 2 = 40

This is an analogy question requiring us to find the relationship between the first pair of numbers (78 and 225) and apply it to the second pair. For 78: If we add the digits (7 + 8), we get 15. If we square this sum (15²), we get 225. For 89: We apply the same logic. Add the digits (8 + 9) to get 17. Then, square this sum (17²). 17 × 17 = 289. Thus, the missing number is 289.

From the statements, we know that there is an intersection between "Letters" and "Pins" (Some letters are pins). However, we do not know the exact relationship between the entire group of Pins and Letters. Scenario A: The group of Pins could be entirely inside the group of Letters. In this case, "All pins are letters" would be true. Scenario B: The group of Pins could partially overlap with Letters, leaving some Pins outside. In this case, "Some pins are not letters" would be true. Since these two conclusions cover all possible logical scenarios (either all are, or some are not), they form a complementary pair. Therefore, the answer is "Either conclusion 1 or 2 follows."

Conclusion I: "Some boxes are not boards." The statements tell us "No pin is a box," but they do not define the relationship between Boxes and Boards. It is possible that all boxes are boards, or no boxes are boards. Since we cannot be certain, this conclusion does not follow. Conclusion II: "No pins are white." The statements link Pins to Boards, and Boards to White. However, there is no rule preventing the "Pin" circle from overlapping with the "White" circle. Pins could be white, or they might not be. Because there is no definite negative relationship established between Pins and White, this conclusion does not follow.

To find the odd one out, we look at the exponential properties of the numbers. 1 = 1⁵ 32 = 2⁵ 243 = 3⁵ All of these numbers are perfect fifth powers of an integer. 256, however, is 4⁴ (or 2⁸). It cannot be written as an integer to the power of 5. Therefore, 256 breaks the pattern found in the other three numbers.

First, we must determine the original schedule based on the puzzle clues provided in the text:

1. Fixed Clue: Dell is distributed on Saturday.

2. Dependent Clue: HP is after Dell. Since Dell is Saturday, HP must be Sunday.

3. Dependent Clue: HP is 3 days after Asus. If HP is Sunday, count back 3 days (Saturday, Friday, Thursday). So, Asus is Thursday.

4. Dependent Clue: Lenovo is "just after" Asus. Since Asus is Thursday, Lenovo must be Friday.

5. Remaining Clues: We have Monday, Tuesday, and Wednesday left. The laptops remaining are HCL, Sony, and Acer.

   • "Acer is just after Sony." This creates a block [Sony, Acer].

   • "HCL is before Acer."

   • To fit HCL before the [Sony, Acer] block in the three available days, the order must be: Monday (HCL), Tuesday (Sony), Wednesday (Acer). Original Schedule: Mon(HCL), Tue(Sony), Wed(Acer), Thu(Asus), Fri(Lenovo), Sat(Dell), Sun(HP).

This question asks for the datum from the original schedule derived above.

1. As established, Asus is on Thursday.

2. The text states "Lenovo laptops are given just after Asus laptops."

3. Therefore, Lenovo is distributed on Friday.

Identify the day for Lenovo: Friday.

We need to analyze the transformation from DANGER (6 letters) to RDEAGN.

• First Half (DAN): The letters D, A, N are moved to the even positions (2nd, 4th, 6th) in order.

  • D → Pos 2

  • A → Pos 4

  • N → Pos 6

• Second Half (GER): The letters G, E, R are moved to the odd positions (5th, 3rd, 1st) in reverse order.

  • R (last letter) → Pos 1

  • E (middle letter) → Pos 3

  • G (first letter) → Pos 5

Apply this to PARALLEL (8 letters):

• First Half (PARA): These 4 letters go to even positions (2, 4, 6, 8) in order.

  • P → 2, A → 4, R → 6, A → 8.

  • Skeleton: _ P _ A _ R _ A

• Second Half (LLEL): These 4 letters go to odd positions (7, 5, 3, 1) in reverse order.

  • Last 'L' → Pos 1

  • 'E' → Pos 3

  • 'L' → Pos 5

  • First 'L' → Pos 7

  • Skeleton: L _ E _ L _ L _

    Combined Result: L P E A L R L A