2
Fill in the blanks to complete the alphabetical series: Z, ?, T, ?, N, ?, H, ?, B
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Solution: Step 1: Convert the known letters to their alphabetical positions (A=1, ..., Z=26).
- Z=26, T=20, N=14, H=8, B=2.
Step 2: Calculate the difference in positions between the known consecutive letters that are separated by one blank.
- From Z (26) to T (20): The total difference is 26 - 20 = 6 positions. Since there is one blank, this difference is split into two steps. So, each step involves a decrement of 6 / 2 = 3 positions.
- From T (20) to N (14): The total difference is 20 - 14 = 6 positions. Each step is a decrement of 3 positions.
- From N (14) to H (8): The total difference is 14 - 8 = 6 positions. Each step is a decrement of 3 positions.
- From H (8) to B (2): The total difference is 8 - 2 = 6 positions. Each step is a decrement of 3 positions.
Step 3: The consistent pattern is a decrement of 3 positions for each step in the series.
Step 4: Fill in the missing terms using this pattern:
- Z (26) - 3 = 23. The 23rd letter is W.
- T (20) - 3 = 17. The 17th letter is Q.
- N (14) - 3 = 11. The 11th letter is K.
- H (8) - 3 = 5. The 5th letter is E.
Step 5: The missing terms are W, Q, K, E.
3
Arrange the following terms in a logical sequence: Plant, Cotton, Yarn, Cloth, Saree.
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Solution: Step 1: Understand the terms given and their logical order
- Plant: Source of cotton
- Cotton: Raw material from the plant
- Yarn: Material made from cotton
- Cloth: Fabric made from yarn
- Saree: Garment made from cloth
Step 2: Determine the logical sequence
The sequence should reflect the progression from source to final product:
1. Plant (source)
2. Cotton (raw material)
3. Yarn (processed material)
4. Cloth (fabric)
5. Saree (final product)
Step 3: Match with given options
The correct sequence is: 2, 4, 1, 5, 3
4
Identify the incorrect number in the sequence: 2, 5, 10, 17, 26, 37, 50, 64.
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Solution: Step 1: Express each term in relation to a perfect square:
- 2 = 1^2 + 1
- 5 = 2^2 + 1
- 10 = 3^2 + 1
- 17 = 4^2 + 1
- 26 = 5^2 + 1
- 37 = 6^2 + 1
- 50 = 7^2 + 1
Step 2: The pattern is (N^2 + 1), where N is a consecutive integer starting from 1.
Step 3: Following this pattern, the next term after 50 should be 8^2 + 1 = 64 + 1 = 65.
Step 4: The given term is 64, which is incorrect.
Step 5: Therefore, 64 is the wrong term.
5
Which number is incorrect in the given sequence: 64, 71, 80, 91, 104, 119, 135, 155?
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Solution: Step 1: Calculate the differences between consecutive terms to identify the pattern.
Step 2: 71 - 64 = 7
Step 3: 80 - 71 = 9
Step 4: 91 - 80 = 11
Step 5: 104 - 91 = 13
Step 6: The differences are 7, 9, 11, 13, which are consecutive odd numbers. The next difference should be 15.
Step 7: So, the expected next term after 104 is 104 + 15 = 119. (Correct)
Step 8: The next difference should be 17. So, the expected next term after 119 is 119 + 17 = 136.
Step 9: The given number at this position is 135, which is incorrect. It should be 136.
Step 10: Verify the subsequent term: If the number were 136, the next difference would be 155 - 136 = 19 (which follows 17 + 2 = 19).
Step 11: Therefore, 135 is the wrong number in the series.
7
Determine the next number in the sequence: 3, 8, 13, 24, 41, ?
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Solution: Step 1: Examine the series: 3, 8, 13, 24, 41, ?
Step 2: Look for a recursive pattern where each term depends on preceding terms and an additional value.
Step 3: Identify the pattern based on the relationship: `Term(n) = Term(n-2) + Term(n-1) + (n-1)`.
Step 4: Verify the pattern for the given terms:
- Term 3 (13) = Term 1 (3) + Term 2 (8) + (3-1) = 3 + 8 + 2 = 13 (Matches)
- Term 4 (24) = Term 2 (8) + Term 3 (13) + (4-1) = 8 + 13 + 3 = 24 (Matches)
- Term 5 (41) = Term 3 (13) + Term 4 (24) + (5-1) = 13 + 24 + 4 = 41 (Matches)
Step 5: Apply the pattern to find the 6th term (T6).
- T6 = Term 4 (24) + Term 5 (41) + (6-1)
- T6 = 24 + 41 + 5
Step 6: Calculate the missing term: 24 + 41 + 5 = 65 + 5 = 70.
Step 7: The missing term is 70.
8
Which number is out of sequence in: 1, 3, 12, 25, 48?
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Solution: Step 1: Analyze the terms by considering pairs of squared numbers:
- 1 = 1^2 - 0^2
- 3 = 2^2 - 1^2
- 12 = 4^2 - 2^2 (which is 16 - 4)
Step 2: Observe the bases of the squares:
- First bases: 1, 2, 4 (this sequence doubles: 1, 1*2, (1*2)*2). The next should be 6.
- Second bases: 0, 1, 2 (this sequence increments by 1). The next should be 3.
Step 3: So, the next term should be 6^2 - 3^2 = 36 - 9 = 27.
Step 4: The given term is 25, which is incorrect.
Step 5: Verify the last term with the pattern: 8^2 - 4^2 = 64 - 16 = 48 (Matches).
Step 6: Therefore, 25 is the wrong term.
12
Complete the sequence: 2, 3, 7, 8, 13, 14, ?
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Solution: Step 1: Observe the given sequence: 2, 3, 7, 8, 13, 14, ?
Step 2: Break the sequence into two separate series:
Series 1: 2, 7, 13
Series 2: 3, 8, 14
Step 3: Analyze Series 1:
- 2 + 5 = 7
- 7 + 6 = 13
- 13 + 7 = 20
Step 4: Analyze Series 2:
- 3 + 5 = 8
- 8 + 6 = 14
Step 5: Identify the pattern: The difference between adjacent numbers increases by 1 in each series.
Step 6: Apply the pattern to find the next number in Series 1:
- The next number in Series 1 is 20.
Step 7: Conclude the missing number in the original sequence is 20.
14
Complete the given series: QAR, RAS, SAT, TAU, _____
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Solution: Step 1: Observe the pattern in the series: QAR, RAS, SAT, TAU, ?
Step 2: Identify the shift in letters: The first letter of each term shifts one position forward in the alphabet (Q to R to S to T).
Step 3: Analyze the second letter: It remains static as 'A'.
Step 4: Examine the third letter: It shifts one position forward in the alphabet (R to S to T to U).
Step 5: Determine the next term: Following the pattern, the next term should start with 'U', have 'A' in the middle, and end with 'V' (as the next letter after 'U').
Step 6: Therefore, the next term in the series is UAV.
15
Complete the series: 2, 8, 14, __, 34, 48, 62, 80, 98
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Solution: Step 1: Examine the given series for patterns: 2, 8, 14, __, 34, 48, 62, 80, 98
Step 2: Calculate the differences between consecutive numbers: 8-2=6, 14-8=6
Step 3: Observe that the differences between consecutive terms are increasing
Step 4: Identify the pattern of differences: +6, +6, +10, +14, +18
Step 5: Apply the pattern to find the missing term: 14 + 10 = 24
Step 6: Verify the pattern with the rest of the series: 24 + 10 = 34, 34 + 14 = 48, and so on
Step 7: Conclude the missing number in the series is 24
16
Complete the sequence: 3, 15, ?, 63, 99, 143
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Solution: Step 1: Analyze the given numbers for an underlying mathematical pattern.
Step 2: Observe that each term is one less than a perfect square:
- 3 = 4 - 1 = 2² - 1
- 15 = 16 - 1 = 4² - 1
- 63 = 64 - 1 = 8² - 1
- 99 = 100 - 1 = 10² - 1
- 143 = 144 - 1 = 12² - 1
Step 3: Identify the pattern of the bases of these squares (the numbers being squared): 2, 4, ?, 8, 10, 12.
Step 4: This sequence of bases is an arithmetic progression of even numbers, increasing by 2 each time. The missing base should be 6.
Step 5: Apply the pattern (n² - 1) with n=6 to find the missing term.
Step 6: Missing term = 6² - 1 = 36 - 1 = 35.
Step 7: The missing term in the sequence is 35.
17
Identify the incorrect number in the given sequence: 11, 5, 20, 12, 40, 26, 74, 54.
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Solution: Step 1: Identify that the given sequence is a combination of two interleaved series.
Step 2: Extract the first series (odd positions): I. 11, 20, 40, 74.
Step 3: Extract the second series (even positions): II. 5, 12, 26, 54.
Step 4: Analyze the pattern in Series I by calculating differences: 20-11=9, 40-20=20, 74-40=34.
Step 5: Recognize the correct pattern for Series I should be based on doubling differences: +9, +18, +36,....
Step 6: Based on this pattern, the third term in Series I should be (20 + 18) = 38.
Step 7: The given term 40 does not fit the pattern.
Step 8: Therefore, 40 is the incorrect term.
19
Order the following human developmental stages into a meaningful chronological sequence: Puberty, Adulthood, Childhood, Infancy, Senescence, Adolescence.
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Solution: Step 1: The earliest stage is 'Infancy'.
Step 2: Followed by 'Childhood'.
Step 3: Then 'Puberty' begins.
Step 4: 'Adolescence' is the stage after puberty.
Step 5: This leads to 'Adulthood'.
Step 6: The final stage is 'Senescence' (old age).
Step 7: The logical sequence is 4, 3, 1, 6, 2, 5.