Sequences & Patterns — Practice Set

Bank code: ARI-SEQ · Section: Arithmetic · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

Constant differences mean arithmetic, constant ratios mean geometric - and every count is (last - first)/step + 1, because you count the terms, not the gaps.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · ARI-SEQ-001 · MCQ · 45s

What is the 10th term of the arithmetic sequence 3, 7, 11, 15, ... ?

(A) 35
(B) 36
(C) 39
(D) 40
(E) 43

Q2 · ARI-SEQ-003 · MCQ · 50s

What is the 5th term of the geometric sequence 2, 6, 18, 54, ... ?

(A) 90
(B) 108
(C) 162
(D) 324
(E) 486

Q3 · ARI-SEQ-007 · MCQ · 55s

What is the sum of all even integers from 2 to 20, inclusive?

(A) 99
(B) 100
(C) 110
(D) 121
(E) 210

Q4 · ARI-SEQ-015 · Numeric Entry · 50s

How many terms are in the arithmetic sequence 5, 8, 11, ..., 56? Enter your answer as a number.

Numeric entry — write the number.

Q5 · ARI-SEQ-010 · Select all that apply · 60s

Which of the following sequences are arithmetic? Select all that apply.

(A) 2, 4, 8, 16, ...
(B) 3, 7, 11, 15, ...
(C) 1, 4, 9, 16, ...
(D) 10, 7, 4, 1, ...
(E) 5, 5, 5, 5, ...

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · ARI-SEQ-029 · MCQ · 70s

In a geometric sequence, the 2nd term is 6 and the 5th term is 162. What is the common ratio?

(A) 2
(B) 3
(C) 6
(D) 9
(E) 27

Q7 · ARI-SEQ-034 · MCQ · 80s

What is the value of (1 + 2 + 3 + ... + 100) - (2 + 4 + 6 + ... + 100)?

(A) 2401
(B) 2500
(C) 2525
(D) 2550
(E) 5050

Q8 · ARI-SEQ-014 · MCQ · 85s

A sequence repeats the block 4, 9, 2 over and over: 4, 9, 2, 4, 9, 2, 4, 9, 2, ... What is the sum of the first 50 terms?

(A) 240
(B) 244
(C) 251
(D) 253
(E) 255

Q9 · ARI-SEQ-017 · Numeric Entry · 80s

In an arithmetic sequence, the 3rd term is 11 and the 7th term is 27. What is the 15th term? Enter your answer as a number.

Numeric entry — write the number.

Q10 · ARI-SEQ-021 · Numeric Entry · 65s

A sequence is defined recursively by a_1 = 2 and a_n = 3a_(n-1) - 1 for every integer n > 1. What is a_4? Enter your answer as a number.

Numeric entry — write the number.

Q11 · ARI-SEQ-031 · Numeric Entry · 85s

How many 3-digit positive integers are divisible by 11? Enter your answer as a number.

Numeric entry — write the number.

Q12 · ARI-SEQ-023 · Select all that apply · 85s

Consider the arithmetic sequence 6, 10, 14, 18, ... , which continues without end. Which of the following numbers are terms of this sequence? Select all that apply.

(A) 50
(B) 62
(C) 78
(D) 100
(E) 200

Q13 · ARI-SEQ-019 · QC · 65s

Sequence A is arithmetic with first term 4 and common difference 15. Sequence B is geometric with first term 4 and common ratio 2.

Column A: The 5th term of sequence A Column B: The 5th term of sequence B

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Level 3 · Hard — 12 questions · ~19 min

where 162+ is won or lost

Q14 · ARI-SEQ-039 · MCQ · 90s

The sum of the first n terms of an arithmetic sequence is S_n = n^2 + 3n for every positive integer n. What is the common difference?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 6

Q15 · ARI-SEQ-042 · MCQ · 90s

Three numbers form an arithmetic sequence. Their sum is 18 and their product is 162. What are the three numbers?

(A) 1, 6, 11
(B) 2, 6, 10
(C) 3, 6, 9
(D) 4, 6, 8
(E) 6, 6, 6

Q16 · ARI-SEQ-048 · MCQ · 85s

An infinite geometric series has first term a, where a > 0, and its sum is exactly 3 times its first term. What is the common ratio?

(A) 1/4
(B) 1/3
(C) 1/2
(D) 2/3
(E) 3/4

Q17 · ARI-SEQ-053 · MCQ · 90s

In an arithmetic sequence, the 6th term is twice the 3rd term, and the 2nd term is 4. What is the first term?

(A) 0
(B) 2
(C) 3
(D) 4
(E) 6

Q18 · ARI-SEQ-054 · MCQ · 95s

An arithmetic sequence has exactly 19 terms. The sum of all 19 terms is 380 and the common difference is 3. What is the largest term?

(A) 20
(B) 44
(C) 47
(D) 50
(E) 74

Q19 · ARI-SEQ-046 · QC · 85s

S is the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... , in which each term is half the term before it.

Column A: S Column B: 1 - 1/2^100

Q20 · ARI-SEQ-051 · QC · 90s

A sequence is defined by a_1 = 1 and a_n = a_(n-1) + n - 1 for every integer n >= 2.

Column A: a_10 Column B: 46

Q21 · ARI-SEQ-061 · QC · 105s

Column A: The number of multiples of 3 from 3 to 300, inclusive, that are not multiples of 5 Column B: The number of integers from 50 to 150, inclusive, that are not multiples of 5

Q22 · ARI-SEQ-043 · Numeric Entry · 105s

What is the value of 1x2 + 2x3 + 3x4 + ... + 10x11? Enter your answer as a number.

Numeric entry — write the number.

Q23 · ARI-SEQ-062 · Numeric Entry · 105s

What is the value of 1/(1x2) + 1/(2x3) + 1/(3x4) + ... + 1/(19x20)? Enter your answer as a number.

Numeric entry — write the number.

Q24 · ARI-SEQ-049 · Select all that apply · 100s

A sequence begins 2, 5, 10, 17, 26, ... and continues by the same rule. Which of the following statements are true? Select all that apply.

(A) The nth term is n^2 + 1
(B) The 10th term is 100
(C) The differences between consecutive terms form an arithmetic sequence with common difference 2
(D) 122 is a term of the sequence
(E) 200 is a term of the sequence

Q25 · ARI-SEQ-044 · Select all that apply · 105s

A geometric sequence has first term 6 and common ratio 3/2. Which of the following statements are true? Select all that apply.

(A) The 4th term is 20.25
(B) The sum of the first 4 terms is 48.75
(C) The infinite series 6 + 9 + 13.5 + ... has a sum of -12
(D) Every term after the first is greater than the term before it
(E) The 10th term is 10 times the 1st term

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · ARI-SEQ-050 · MCQ · 115s

In a geometric sequence every term is positive. The sum of the first two terms is 12 and the sum of the first four terms is 36. What is the sum of the first six terms?

(A) 48
(B) 60
(C) 72
(D) 84
(E) 108

Q27 · ARI-SEQ-041 · QC · 115s

An infinite geometric sequence has first term a > 0 and common ratio r with 0 < r < 1. S denotes the sum of the entire infinite series.

Column A: The sum of the first 10 terms of the sequence Column B: S x (1 - r^10)

Q28 · ARI-SEQ-059 · QC · 120s

An arithmetic sequence has first term a and common difference d, where a > 0 and d > 0, and n is a positive integer.

Column A: (the sum of the first 2n terms) divided by (the sum of the first n terms) Column B: 4

Q29 · ARI-SEQ-056 · Numeric Entry · 125s

What is the value of 1^2 - 2^2 + 3^2 - 4^2 + ... + 99^2 - 100^2? Enter your answer as a number.

Numeric entry — write the number.

Q30 · ARI-SEQ-057 · Select all that apply · 110s

An infinite geometric series has first term a, where a is not 0, and common ratio r. Which of the following conditions, on its own, guarantees that the series has a finite sum? Select all that apply.

(A) |r| < 1
(B) r < 1
(C) 0 < r < 1
(D) -1 < r < 0
(E) a > 0 and r > 0

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