46. Logical or arithmetic patterns problems

Logical or arithmetic patterns problems involve identifying and extending sequences or patterns based on numerical or logical rules. These problems often require recognizing relationships between terms (e.g., arithmetic, geometric, or other patterns) to determine missing terms, the next term, or the nth term in a sequence. They are common in mathematics, logic puzzles, and aptitude tests, testing pattern recognition and problem-solving skills.

Below are 10 logical or arithmetic patterns problems with step-by-step solutions and detailed explanations. Each problem includes the problem statement, solution procedure, verification, and an explanation of the concept. The problems are diverse, covering arithmetic sequences, geometric sequences, recursive patterns, and logical number patterns, ensuring they are distinct from previous problem types (e.g., age or estimation problems). They focus on numerical sequences and avoid unrelated contexts (e.g., alcohol or mixing solutions).

Problem 1

Find the next term in the sequence: 3, 7, 11, 15, 19.

Step-by-Step Solution:

Identify the pattern: List the terms and check differences between consecutive terms:
7 - 3 = 4, \quad 11 - 7 = 4, \quad 15 - 11 = 4, \quad 19 - 15 = 4
The differences are constant (4), indicating an arithmetic sequence with a common difference of 4.
Extend the sequence: Add 4 to the last term (19):
19 + 4 = 23
Verify: The sequence is ( 3, 7, 11, 15, 19, 23 ). Differences:
7 - 3 = 4
,
11 - 7 = 4
, ...,
23 - 19 = 4
. The pattern holds.
Answer: The next term is 23.

Explanation: This is a classic arithmetic sequence where each term increases by a fixed amount (common difference = 4). Recognizing the constant difference allows us to predict the next term by adding 4 to the last term. Arithmetic sequences are defined by

a_n = a_1 + (n-1)d

, where ( d ) is the common difference.

Problem 2

Determine the 10th term in the sequence: 2, 5, 8, 11, ...

Step-by-Step Solution:

Identify the pattern: Check differences:
5 - 2 = 3, \quad 8 - 5 = 3, \quad 11 - 8 = 3
The common difference is 3, indicating an arithmetic sequence.
Use the arithmetic sequence formula: The nth term of an arithmetic sequence is:
a_n = a_1 + (n-1)d
where
a_1 = 2
,
d = 3
, and we need the 10th term (
n = 10
):
a_{10} = 2 + (10-1) \cdot 3 = 2 + 9 \cdot 3 = 2 + 27 = 29
Verify: List terms: ( 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 ). The 10th term is 29, and differences are consistently 3.
Answer: The 10th term is 29.

Explanation: This arithmetic sequence has a common difference of 3. The formula efficiently finds the nth term without listing all terms, making it ideal for large ( n ). The verification confirms the pattern by checking intermediate terms.

Problem 3

Find the next term in the sequence: 1, 2, 4, 8, 16.

Step-by-Step Solution:

Identify the pattern: Check ratios between consecutive terms:
\frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{8}{4} = 2, \quad \frac{16}{8} = 2
Each term is multiplied by 2, indicating a geometric sequence with a common ratio of 2.
Extend the sequence: Multiply the last term (16) by 2:
16 \cdot 2 = 32
Verify: Sequence: ( 1, 2, 4, 8, 16, 32 ). Ratios:
\frac{2}{1} = 2
, ...,
\frac{32}{16} = 2
. The pattern holds.
Answer: The next term is 32.

Explanation: This is a geometric sequence where each term is obtained by multiplying the previous term by a fixed ratio (2). Geometric sequences are defined by

a_n = a_1 \cdot r^{n-1}

, and here, the simple multiplication extends the sequence.

Problem 4

Find the 8th term in the sequence: 3, 9, 27, 81, ...

Step-by-Step Solution:

Identify the pattern: Check ratios:
\frac{9}{3} = 3, \quad \frac{27}{9} = 3, \quad \frac{81}{27} = 3
The common ratio is 3, indicating a geometric sequence.
Use the geometric sequence formula: The nth term is:
a_n = a_1 \cdot r^{n-1}
where
a_1 = 3
,
r = 3
, and we need the 8th term (
n = 8
):
a_8 = 3 \cdot 3^{8-1} = 3 \cdot 3^7 = 3 \cdot 2,187 = 6,561
Verify: List terms: ( 3, 9, 27, 81, 243, 729, 2,187, 6,561 ). The 8th term is 6,561, and ratios are consistently 3 (e.g.,
\frac{6,561}{2,187} = 3
).
Answer: The 8th term is 6,561.

Explanation: This geometric sequence has a common ratio of 3. The formula allows us to compute the 8th term directly, avoiding manual multiplication for each term. The verification confirms the exponential growth pattern.

Problem 5

Find the next term in the sequence: 1, 4, 9, 16, 25.

Step-by-Step Solution:

Identify the pattern: Recognize the terms:
1 = 1^2, \quad 4 = 2^2, \quad 9 = 3^2, \quad 16 = 4^2, \quad 25 = 5^2
The terms are perfect squares of consecutive integers (
n^2
).
Extend the sequence: The next term corresponds to
n = 6
:
6^2 = 36
Verify: Sequence: ( 1, 4, 9, 16, 25, 36 ). Terms are
1^2, 2^2, 3^2, 4^2, 5^2, 6^2
, and the pattern holds.
Answer: The next term is 36.

Explanation: This sequence follows a quadratic pattern where each term is the square of its position number. Recognizing the perfect squares is key, and extending the pattern is straightforward by squaring the next integer.

Problem 6

Determine the next term in the sequence: 2, 5, 11, 23, 47.

Step-by-Step Solution:

Identify the pattern: Check differences:
5 - 2 = 3, \quad 11 - 5 = 6, \quad 23 - 11 = 12, \quad 47 - 23 = 24
Differences: ( 3, 6, 12, 24 ). Notice the pattern in differences:
6 \div 3 = 2, \quad 12 \div 6 = 2, \quad 24 \div 12 = 2
Each difference is doubled, so the next difference is
24 \cdot 2 = 48
.
Extend the sequence: Add the next difference to the last term:
47 + 48 = 95
Verify: Sequence: ( 2, 5, 11, 23, 47, 95 ). Differences: ( 3, 6, 12, 24, 48 ). Each difference doubles, confirming the pattern.
Answer: The next term is 95.

Explanation: This is a recursive sequence where the differences form a geometric pattern (doubling). The pattern in differences allows us to predict the next term by calculating the next difference and adding it to the last term.

Problem 7

Find the 7th term in the sequence: 1, 1, 2, 3, 5, 8, ...

Step-by-Step Solution:

Identify the pattern: Observe the terms:
1 + 1 = 2, \quad 1 + 2 = 3, \quad 2 + 3 = 5, \quad 3 + 5 = 8
Each term is the sum of the two preceding terms, indicating a Fibonacci-like sequence starting with 1, 1.
Extend the sequence:
- 6th term = 8
- 7th term:
  5 + 8 = 13
Verify: Sequence: ( 1, 1, 2, 3, 5, 8, 13 ). Check:
5 + 8 = 13
, consistent with the pattern.
Answer: The 7th term is 13.

Explanation: This is a Fibonacci sequence where each term is the sum of the two previous terms. The recursive nature requires listing terms up to the desired position, making it a classic example of a recursive pattern.

Problem 8

Find the next term in the sequence: 4, 7, 13, 22, 34.

Step-by-Step Solution:

Identify the pattern: Check differences:
7 - 4 = 3, \quad 13 - 7 = 6, \quad 22 - 13 = 9, \quad 34 - 22 = 12
Differences: ( 3, 6, 9, 12 ). The differences increase by 3 each time, forming an arithmetic sequence of differences.
Extend the differences: Next difference =
12 + 3 = 15
.
Extend the sequence: Add the next difference to the last term:
34 + 15 = 49
Verify: Sequence: ( 4, 7, 13, 22, 34, 49 ). Differences: ( 3, 6, 9, 12, 15 ). The differences increase by 3, confirming the pattern.
Answer: The next term is 49.

Explanation: This sequence has an arithmetic pattern in its differences, which themselves form an arithmetic sequence (common difference = 3). The next term is found by adding the next difference, illustrating a second-order pattern.

Problem 9

Determine the 6th term in the sequence: 2, 6, 12, 20, 30, ...

Step-by-Step Solution:

Identify the pattern: Check differences:
6 - 2 = 4, \quad 12 - 6 = 6, \quad 20 - 12 = 8, \quad 30 - 20 = 10
Differences: ( 4, 6, 8, 10 ). The differences increase by 2 each time.
Generalize the sequence: The nth term appears related to a pattern. Notice the terms resemble
n(n+1)
:
- 1 \cdot 2 = 2
- 2 \cdot 3 = 6
- 3 \cdot 4 = 12
- 4 \cdot 5 = 20
- 5 \cdot 6 = 30
  The nth term is
  n(n+1)
  .
Find the 6th term:
a_6 = 6 \cdot (6+1) = 6 \cdot 7 = 42
Verify: Sequence: ( 2, 6, 12, 20, 30, 42 ). Differences: ( 4, 6, 8, 10, 12 ). The differences increase by 2, and terms match
n(n+1)
.
Answer: The 6th term is 42.

Explanation: This sequence follows a quadratic pattern where the nth term is

n(n+1)

. The differences confirm the pattern (increasing by 2), and the formula provides a direct way to find any term, making it efficient for the 6th term.

Problem 10

Find the next term in the sequence: 1, 3, 6, 10, 15.

Step-by-Step Solution:

Identify the pattern: Check differences:
3 - 1 = 2, \quad 6 - 3 = 3, \quad 10 - 6 = 4, \quad 15 - 10 = 5
Differences: ( 2, 3, 4, 5 ). The differences increase by 1 each time.
Recognize the pattern: The terms are triangular numbers, given by:
a_n = \frac{n(n+1)}{2}
- n=1
  :
  \frac{1 \cdot 2}{2} = 1
- n=2
  :
  \frac{2 \cdot 3}{2} = 3
- n=3
  :
  \frac{3 \cdot 4}{2} = 6
- n=4
  :
  \frac{4 \cdot 5}{2} = 10
- n=5
  :
  \frac{5 \cdot 6}{2} = 15
Extend the sequence: For
n=6
:
a_6 = \frac{6 \cdot 7}{2} = \frac{42}{2} = 21
Verify: Sequence: ( 1, 3, 6, 10, 15, 21 ). Differences: ( 2, 3, 4, 5, 6 ). The differences increase by 1, and terms match the triangular number formula.
Answer: The next term is 21.

Explanation: This sequence consists of triangular numbers, where each term is the sum of the first ( n ) natural numbers (

1 + 2 + \cdots + n

). The formula

\frac{n(n+1)}{2}

provides a direct way to find the next term, and the differences confirm the pattern.

General Notes on Logical or Arithmetic Patterns Problems

Key Concept: These problems require identifying the rule governing a sequence to predict future terms or find specific terms. Common patterns include arithmetic (constant difference), geometric (constant ratio), recursive (e.g., Fibonacci), or functional (e.g., squares, triangular numbers).
Key Steps:
1. List the terms and calculate differences or ratios to identify the pattern.
2. Hypothesize a rule (e.g., arithmetic, geometric, or a formula like
  n^2
  ).
3. Apply the rule to find the requested term (next term or nth term).
4. Verify by checking if the pattern holds for all given terms and the new term.
Common Patterns:
- Arithmetic:
  a_n = a_1 + (n-1)d
  , where ( d ) is the common difference.
- Geometric:
  a_n = a_1 \cdot r^{n-1}
  , where ( r ) is the common ratio.
- Fibonacci-like:
  a_n = a_{n-1} + a_{n-2}
  .
- Quadratic/Functional: Terms like
  n^2
  ,
  n(n+1)
  , or
  \frac{n(n+1)}{2}
  .
- Differences Pattern: Differences may form their own sequence (e.g., arithmetic or geometric).
Challenges:
- Distinguish between arithmetic, geometric, or other patterns by checking differences or ratios.
- For complex patterns, consider differences of differences or functional forms (e.g., squares, triangular numbers).
- Ensure the pattern is consistent across all terms, as some sequences may have multiple possible rules.
Applications: These problems appear in mathematics (sequence analysis), computer science (algorithms), aptitude tests, and puzzles, requiring logical reasoning and pattern recognition.
Verification: Always verify by applying the identified rule to all given terms and the predicted term, checking differences, ratios, or formula consistency.

These problems showcase a variety of patterns, from simple arithmetic and geometric sequences to recursive and quadratic patterns, demonstrating the breadth of logical and arithmetic sequence problems.

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46. Logical or arithmetic patterns problems

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