6.3 Sequences & Mathematical Patterns

6.3 Sequences & Mathematical Patterns

Sequences and mathematical patterns are a standard component of the GAT Quantitative section. Qiyas uses sequence problems to assess your logical reasoning, visual-spatial pattern recognition, and capacity to handle numerical progressions. While some sequences follow standard arithmetic or geometric definitions, others are non-standard patterns that require logical heuristics to crack. Understanding the mechanics of both standard progressions and non-standard sequences is the key to solving these questions quickly and accurately.

Arithmetic Sequences (Progressions)

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d). The sequence: 3, 7, 11, 15, 19... has a common difference of d = 4.

Key Arithmetic Formulas

1. n-th Term Formula: a_n = a_1 + (n - 1)d Where a_n is the n-th term, a_1 is the first term, and d is the common difference.
2. Number of Terms (n) in a finite range: n = ((a_n - a_1) / d) + 1 This formula is essential when you need to count how many multiples of a number exist in a range.
3. Sum of an Arithmetic Sequence (S_n): S_n = n * (a_1 + a_n) / 2 The sum is the number of terms multiplied by the average of the first and last terms.

Worked Example: Finding the n-th Term and Sum

Find the 20th term and the sum of the first 20 terms of the sequence: 5, 9, 13, 17...

• Step 1: Identify the components. First term (a_1) = 5 Common difference (d) = 9 - 5 = 4 Number of terms (n) = 20
• Step 2: Apply the n-th term formula: a_20 = 5 + (20 - 1)4 = 5 + 19 * 4 = 5 + 76 = 81
• Step 3: Apply the sum formula: S_20 = 20 * (5 + 81) / 2 = 10 * 86 = 860 The 20th term is 81 and the sum is 860.

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Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). The sequence: 2, 6, 18, 54... has a common ratio of r = 3.

Key Geometric Formulas

1. n-th Term Formula: a_n = a_1 * r^(n-1) Where a_n is the n-th term, a_1 is the first term, and r is the common ratio.

Worked Example: Finding a Term in a Geometric Sequence

Find the 6th term of the geometric sequence: 3, 6, 12, 24...

• Step 1: Identify the components. First term (a_1) = 3 Common ratio (r) = 6 / 3 = 2 Term to find (n) = 6
• Step 2: Apply the formula: a_6 = 3 * 2^(6 - 1) = 3 * 2^5 = 3 * 32 = 96 The 6th term is 96.

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Non-Standard GAT Sequences and Pattern Heuristics

GAT is famous for sequences that are neither purely arithmetic nor geometric. To solve these, you must inspect the terms and test the following common logic patterns:

1. Alternating (Interleaved) Sequences

An alternating sequence merges two independent sequences. The odd-positioned terms follow one rule, while the even-positioned terms follow a different rule.

• Example: 2, 20, 4, 17, 6, 14, 8...
• Odd terms: 2, 4, 6, 8 (adding 2)
• Even terms: 20, 17, 14 (subtracting 3)
• The next term in the sequence would be the next even term: 14 - 3 = 11.

2. Progressive Differences

The difference between terms is not constant, but the differences themselves form an arithmetic or geometric sequence.

• Example: 3, 4, 7, 12, 19...
• Differences: +1, +3, +5, +7... (adding consecutive odd numbers)
• The next difference will be +9, making the next term: 19 + 9 = 28.

3. Fibonacci-Type Sequences

Each term is the sum of the preceding two terms: a_n = a_(n-1) + a_(n-2).

• Example: 1, 2, 3, 5, 8, 13, 21...
• Notice: 1+2=3, 2+3=5, 3+5=8, etc.
• The next term is 13 + 21 = 34.

4. Exponential and Squaring Patterns

The terms are squares, cubes, or powers of numbers, or are close to them (n^2 + 1, n^2 - 1).

• Example: 2, 5, 10, 17, 26...
• Notice: (1^2 + 1) = 2, (2^2 + 1) = 5, (3^2 + 1) = 10, (4^2 + 1) = 17, (5^2 + 1) = 26.
• The next term is 6^2 + 1 = 37.

Sequences Classification Summary

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Gauss Summation: Sum of Consecutive Integers

A classic GAT question asks for the sum of a series of consecutive integers. The formula for the sum of the first N positive integers is: Sum = N * (N + 1) / 2 For example, the sum of integers from 1 to 100 is: 100 * 101 / 2 = 50 * 101 = 5050.

If the range does not start at 1 (e.g., the sum of integers from 10 to 50), use the general arithmetic sum formula: Sum = n * (First Term + Last Term) / 2 Where n is the number of terms. Remember the inclusive counting rule: Number of terms (n) = Last Term - First Term + 1

For 10 to 50: n = 50 - 10 + 1 = 41 terms Sum = 41 * (10 + 50) / 2 = 41 * 30 = 1230.

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Common GAT Sequence Traps

1. The Range Count Omission: When counting integers in a range (such as "integers between 10 and 20 inclusive"), forgetting to add 1 is a major trap. 20 - 10 = 10, but there are 11 numbers in that range. Always add 1.
2. First-Term Inaccuracy: In alternating sequences, make sure you trace the correct sub-sequence. It is easy to accidentally apply the rule of the even terms to an odd-positioned question or vice versa.
3. Premature Extrapolations: Do not assume a pattern based on only the first two terms. The sequence 2, 4... could be arithmetic (+2 giving 6) or geometric (*2 giving 8). Look at the third term to confirm the pattern before calculating.