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17 Apr 2025, 04:22
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GMAT Question Patterns: Sequences
Table of Contents:
• Arithmetic Sequences
• Number Strings
• Recursive Sequences
• Position-Based Rules
• Theory
• Questions
Pattern #1: Arithmetic Sequences
- Each term increases or decreases by a fixed value.
Example 1.1 (Difficulty: 505-555 Level )
If 4, 7, 10, and 13 are the first four terms of an arithmetic progression, which term is 124 ?
A. The 30th
B. The 31st
C. The 32nd
D. The 40th
E. The 41st
Question Discussion.
Pattern #2: Number Strings
- Involves a string of digits created by joining together numbers that follow a pattern.
Example 2.1 (Difficulty: 655-705 Level)
The string of digits 3691215...300 is formed by merging together the decimal representations of the first 100 positive integer multiples of 3. Counting from the left, what is the 72nd digit of this string of digits?
A. 0
B. 1
C. 6
D. 8
E. 9
Question Discussion.
Example 2.2 (Difficulty: 655-705 Level)
The string of digits 135791113...999 is formed by merging together the decimal representations of the odd integers from 1 through 999. Counting from left, what is the 110th digit of this string of digits?
A. 0
B. 3
C. 5
D. 7
E. 9
Question Discussion.
Pattern #3: Recursive Sequences
- Involves sequences where each term is defined based on one or more previous terms using a fixed rule (e.g., aₙ = aₙ−1 + aₙ−2).
Example 3.1 (Difficulty: Sub 505 Level)
In the sequence \(a_1, \ a_2, \ a_3, \ ..., \ a_n, \ ...\), the terms \(a_i\), \(a_{i+1}\), and \(a_{i+2}\) satisfy the equation \(a_{i+2} = 2a_{i+1} + 3a_i\) for all positive integers i. If \(a_3 = 9\) and \(a_5 = 51\), what is the value of \(a_4\)?
A. 12
B. 30
C. 72
D. 129
E. 171
Question Discussion.
Example 3.2 (Difficulty: Sub 505 Level)
In an economics model, \(A_t\), is the annual income, in millions of dollars, of an economic system after t years for t = 0, 1, 2, ... If At satisfies the relation \(A_t = \frac{3}{2}A_{t - 1} - A_{t - 2} + 1\), where \(A_0 = 0\) and \(A_1 = 1\), what is the annual income, in millions of dollars, after 3 years?
A. 0
B. 1.5
C. 2.5
D. 3.75
E. 4.125
Question Discussion.
Example 3.3 (Difficulty: Sub 505 Level)
The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_1 = 3\), \(a_2 = 5\), and \(a_n = a_{n-1} - a_{n-2}\) for all integers \(n ≥ 3\). What is the sum of the first 100 terms of the sequence?
(A) 2
(B) 3
(C) 7
(D) 8
(E) 10
Question Discussion.
Example 3.4 (Difficulty: Sub 505 Level)
In a certain sequence the first term is 1, and the second term is 2. After the second term, each term can be obtained by subtracting from the previous term the term before it. For example, the third term is 2-1=1. How many different values are possible for the sum of the first n terms ?
A. 2
B. 3
C. 4
D. 5
E. More than 5
Question Discussion.
Example 3.5 (Difficulty: Sub 505 Level)
The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_n=a_{n−1}-a_{n−2}\) for all integers \(n\geq{3}\). If \(a_1=-1\) and \(a_2=1\), what is the sum of the first 1,000 terms of the sequence?
A. -1
B. 0
C. 2
D. 3
E. 6
Question Discussion.
Pattern #4: Position-Based Rules
- Involves sequences where each term depends directly on its position (e.g., aₙ = (–1)n × n or aₙ = n2 + 1), without referring to previous terms.
Example 4.1 (Difficulty: 505-555 Level )
The sequence \(a_0\), \(a_1\), \(a_2\), ..., \(a_n\), ..., is such that \(a_0 = 1\) and \(a_{i+1} =ka_i\) for all positive integers \(i\), where \(k\) is a positive constant. What is the value of \(\frac{(a_{100}+a_{102})}{a_{101}}\) in terms of k?
A. \(k^2\)
B. \(k^{101}\)
C. \(k + 1\)
D. \(k + \frac{1}{k}\)
E. \(k^2 + \frac{1}{k^2} \)
Question Discussion.
Example 4.2 (Difficulty: 605-655 Level)
For each positive integer k, let \(a_k= 7k\). Which of the following is the greatest value of n such that \(10^{n}\) divides \((a_1)(a_2)(a_3)...(a_{28})\)?
A. 7
B. 6
C. 5
D. 4
E. 3
Question Discussion.
Example 4.3 (Difficulty: 655-705 Level)
If \(a_n = \frac{(-1)^n}{n^2}\) and \(b_n = \frac{(-1)^{n + 1}}{n(n + 1)}\) for all positive integers n, which of the following must be true?
A. \(a_1 > b_1\)
B. \(a_2 > b_1\)
C. \(a_3 > b_2\)
D. \(a_4 > b_3\)
E. \(a_5 > b_1\)
Question Discussion.
Example 4.4 (Difficulty: 655-705 Level)
For each positive integer n, let \(a_n = n(-1)^n(x-1)\). If the sum of \(a_1\) through \(a_{20}\) is equal to 60, what is the value of x?
(A) -5
(B) -2
(C) 2
(D) 4
(E) 7
Question Discussion.
Theory:
• Sequences Made Easy - All in One Topic!
• Math : Sequences & Progressions
Questions:
• Hard Questions
• Medium Questions
• Easy Questions
Table of Contents:
• Arithmetic Sequences
• Number Strings
• Recursive Sequences
• Position-Based Rules
• Theory
• Questions
Pattern #1: Arithmetic Sequences
- Each term increases or decreases by a fixed value.
Example 1.1 (Difficulty: 505-555 Level )
If 4, 7, 10, and 13 are the first four terms of an arithmetic progression, which term is 124 ?
A. The 30th
B. The 31st
C. The 32nd
D. The 40th
E. The 41st
Question Discussion.
Pattern #2: Number Strings
- Involves a string of digits created by joining together numbers that follow a pattern.
Example 2.1 (Difficulty: 655-705 Level)
The string of digits 3691215...300 is formed by merging together the decimal representations of the first 100 positive integer multiples of 3. Counting from the left, what is the 72nd digit of this string of digits?
A. 0
B. 1
C. 6
D. 8
E. 9
Question Discussion.
Example 2.2 (Difficulty: 655-705 Level)
The string of digits 135791113...999 is formed by merging together the decimal representations of the odd integers from 1 through 999. Counting from left, what is the 110th digit of this string of digits?
A. 0
B. 3
C. 5
D. 7
E. 9
Question Discussion.
Pattern #3: Recursive Sequences
- Involves sequences where each term is defined based on one or more previous terms using a fixed rule (e.g., aₙ = aₙ−1 + aₙ−2).
Example 3.1 (Difficulty: Sub 505 Level)
In the sequence \(a_1, \ a_2, \ a_3, \ ..., \ a_n, \ ...\), the terms \(a_i\), \(a_{i+1}\), and \(a_{i+2}\) satisfy the equation \(a_{i+2} = 2a_{i+1} + 3a_i\) for all positive integers i. If \(a_3 = 9\) and \(a_5 = 51\), what is the value of \(a_4\)?
A. 12
B. 30
C. 72
D. 129
E. 171
Question Discussion.
Example 3.2 (Difficulty: Sub 505 Level)
In an economics model, \(A_t\), is the annual income, in millions of dollars, of an economic system after t years for t = 0, 1, 2, ... If At satisfies the relation \(A_t = \frac{3}{2}A_{t - 1} - A_{t - 2} + 1\), where \(A_0 = 0\) and \(A_1 = 1\), what is the annual income, in millions of dollars, after 3 years?
A. 0
B. 1.5
C. 2.5
D. 3.75
E. 4.125
Question Discussion.
Example 3.3 (Difficulty: Sub 505 Level)
The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_1 = 3\), \(a_2 = 5\), and \(a_n = a_{n-1} - a_{n-2}\) for all integers \(n ≥ 3\). What is the sum of the first 100 terms of the sequence?
(A) 2
(B) 3
(C) 7
(D) 8
(E) 10
Question Discussion.
Example 3.4 (Difficulty: Sub 505 Level)
In a certain sequence the first term is 1, and the second term is 2. After the second term, each term can be obtained by subtracting from the previous term the term before it. For example, the third term is 2-1=1. How many different values are possible for the sum of the first n terms ?
A. 2
B. 3
C. 4
D. 5
E. More than 5
Question Discussion.
Example 3.5 (Difficulty: Sub 505 Level)
The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_n=a_{n−1}-a_{n−2}\) for all integers \(n\geq{3}\). If \(a_1=-1\) and \(a_2=1\), what is the sum of the first 1,000 terms of the sequence?
A. -1
B. 0
C. 2
D. 3
E. 6
Question Discussion.
Pattern #4: Position-Based Rules
- Involves sequences where each term depends directly on its position (e.g., aₙ = (–1)n × n or aₙ = n2 + 1), without referring to previous terms.
Example 4.1 (Difficulty: 505-555 Level )
The sequence \(a_0\), \(a_1\), \(a_2\), ..., \(a_n\), ..., is such that \(a_0 = 1\) and \(a_{i+1} =ka_i\) for all positive integers \(i\), where \(k\) is a positive constant. What is the value of \(\frac{(a_{100}+a_{102})}{a_{101}}\) in terms of k?
A. \(k^2\)
B. \(k^{101}\)
C. \(k + 1\)
D. \(k + \frac{1}{k}\)
E. \(k^2 + \frac{1}{k^2} \)
Question Discussion.
Example 4.2 (Difficulty: 605-655 Level)
For each positive integer k, let \(a_k= 7k\). Which of the following is the greatest value of n such that \(10^{n}\) divides \((a_1)(a_2)(a_3)...(a_{28})\)?
A. 7
B. 6
C. 5
D. 4
E. 3
Question Discussion.
Example 4.3 (Difficulty: 655-705 Level)
If \(a_n = \frac{(-1)^n}{n^2}\) and \(b_n = \frac{(-1)^{n + 1}}{n(n + 1)}\) for all positive integers n, which of the following must be true?
A. \(a_1 > b_1\)
B. \(a_2 > b_1\)
C. \(a_3 > b_2\)
D. \(a_4 > b_3\)
E. \(a_5 > b_1\)
Question Discussion.
Example 4.4 (Difficulty: 655-705 Level)
For each positive integer n, let \(a_n = n(-1)^n(x-1)\). If the sum of \(a_1\) through \(a_{20}\) is equal to 60, what is the value of x?
(A) -5
(B) -2
(C) 2
(D) 4
(E) 7
Question Discussion.
Theory:
• Sequences Made Easy - All in One Topic!
• Math : Sequences & Progressions
Questions:
• Hard Questions
• Medium Questions
• Easy Questions
17 Apr 2025, 04:27
Kudos
Bookmarks
The whole collection:
- GMAT Question Patterns: Work and Rate Problems
- GMAT Question Patterns: Arithmetic
- GMAT Question Patterns: Absolute Value
- GMAT Question Patterns: Distance and Speed
- GMAT Question Patterns: Remainders
- GMAT Question Patterns: Algebra
- GMAT Question Patterns: Overlapping Sets
- GMAT Question Patterns: Sequences
23 Apr 2025, 06:44
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The new pattern topic: GMAT Question Patterns: Functions and Custom Characters.
