A sequence is an ordered list of numbers following a specific pattern, denoted as \(a_1, a_2, a_3, \dots\). It can be finite or infinite.
Types of Sequences:
Arithmetic Sequence: Each term increases or decreases by a constant difference, \(d\). Formula: \(a_n = a_1 + (n-1)d\).
Geometric Sequence: Each term is multiplied by a constant ratio, \(r\). Formula: \(a_n = a_1 \cdot r^{n-1}\).
Harmonic Sequence: Reciprocals of an arithmetic sequence, e.g., \(1, \frac{1}{2}, \frac{1}{3}, \dots\).
A series is the sum of the terms of a sequence, written as \(s = a_1 + a_2 + a_3 + \dots\).
Types of Series:
Arithmetic Series: Sum of an arithmetic sequence. Sum formula: \(s_n = \frac{n}{2} (a_1 + a_n)\) or \(s_n = \frac{n}{2} [2a_1 + (n-1)d]\).
Geometric Series: Sum of a geometric sequence. Sum of first \(n\) terms: \(s_n = a_1 \frac{1-r^n}{1-r}\) (for \(r \neq 1\)). Infinite sum: \(s = \frac{a_1}{1-r}\) if \(|r| < 1\).
Other Series: Harmonic series (\(1 + \frac{1}{2} + \frac{1}{3} + \dots\)) diverges; telescoping series cancel terms.
Key Concepts:
Convergence: A series converges if its sum approaches a finite limit (e.g., infinite geometric series with \(|r| < 1\)).
Divergence: A series diverges if the sum is infinite or undefined.
Tests for Series: Use ratio test, root test, or integral test to determine convergence.
Table of contents
- Part 1: OnlineExamMaker AI quiz maker – Make a free quiz in minutes
- Part 2: 20 sequences and series quiz questions & answers
- Part 3: AI Question Generator – Automatically create questions for your next assessment
Part 1: OnlineExamMaker AI quiz maker – Make a free quiz in minutes
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Part 2: 20 sequences and series quiz questions & answers
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1. What is the next term in the arithmetic sequence: 3, 7, 11, 15, …?
A) 18
B) 19
C) 20
D) 21
Answer: B) 19
Explanation: The common difference is 4 (7 – 3 = 4), so add 4 to 15 to get 19.
2. Find the sum of the first 5 terms of the arithmetic sequence: 2, 5, 8, 11, 14.
A) 30
B) 35
C) 40
D) 45
Answer: C) 40
Explanation: Use the formula for the sum of an arithmetic series: S_n = n/2 * (a_1 + a_n). Here, n=5, a_1=2, a_5=14, so S_5 = 5/2 * (2 + 14) = 5/2 * 16 = 40.
3. In the geometric sequence: 5, 10, 20, 40, …, what is the 6th term?
A) 80
B) 100
C) 120
D) 80
Answer: A) 80
Explanation: The common ratio is 2 (10/5=2), so the 6th term is 40 * 2 = 80.
4. What is the sum of the first 4 terms of the geometric sequence: 3, 6, 12, 24?
A) 45
B) 40
C) 35
D) 30
Answer: A) 45
Explanation: Use the formula S_n = a * (1 – r^n) / (1 – r). Here, a=3, r=2, n=4, so S_4 = 3 * (1 – 2^4) / (1 – 2) = 3 * (1 – 16) / (-1) = 3 * 15 = 45.
5. Determine if the series ∑(1/2)^n from n=1 to infinity converges.
A) Yes
B) No
C) Only for even n
D) Only for odd n
Answer: A) Yes
Explanation: This is an infinite geometric series with |r| = 1/2 < 1, so it converges to a / (1 - r) = (1/2) / (1 - 1/2) = 1.
6. Find the 10th term of the arithmetic sequence where the first term is 4 and the common difference is 3.
A) 28
B) 31
C) 34
D) 37
Answer: B) 31
Explanation: Use the formula a_n = a_1 + (n-1)d. Here, a_10 = 4 + (10-1)*3 = 4 + 9*3 = 4 + 27 = 31.
7. In the geometric sequence: 1, 1/2, 1/4, 1/8, …, what is the sum of the infinite series?
A) 1
B) 2
C) 1.5
D) 2.5
Answer: B) 2
Explanation: For an infinite geometric series with |r| < 1, sum = a / (1 - r). Here, a=1, r=1/2, so sum = 1 / (1 - 1/2) = 1 / 0.5 = 2.
8. What is the common ratio of the geometric sequence: 2, 6, 18, 54?
A) 2
B) 3
C) 4
D) 5
Answer: B) 3
Explanation: Divide the second term by the first: 6/2 = 3, and 18/6 = 3, so the common ratio is 3.
9. Find the sum of the first 6 terms of the arithmetic sequence: -2, 0, 2, 4, 6, 8.
A) 24
B) 28
C) 30
D) 32
Answer: C) 30
Explanation: S_n = n/2 * (a_1 + a_n). Here, n=6, a_1=-2, a_6=8, so S_6 = 6/2 * (-2 + 8) = 3 * 6 = 18.
10. Which of the following is a geometric sequence?
A) 1, 3, 5, 7
B) 2, 4, 8, 16
C) 5, 10, 15, 20
D) 1, 2, 4, 7
Answer: B) 2, 4, 8, 16
Explanation: Each term is multiplied by 2, indicating a constant ratio of 2.
11. What is the next term in the sequence: 1, 1, 2, 3, 5, 8, …?
A) 10
B) 11
C) 12
D) 13
Answer: D) 13
Explanation: This is the Fibonacci sequence, where each term is the sum of the two preceding ones: 5 + 8 = 13.
12. Find the sum of the geometric series: 1 + 3 + 9 + 27 for 4 terms.
A) 40
B) 39
C) 41
D) 38
Answer: A) 40
Explanation: S_n = a * (1 – r^n) / (1 – r). Here, a=1, r=3, n=4, so S_4 = 1 * (1 – 3^4) / (1 – 3) = (1 – 81) / (-2) = (-80) / (-2) = 40.
13. In an arithmetic sequence, the 3rd term is 10 and the 5th term is 16. What is the first term?
A) 4
B) 5
C) 6
D) 7
Answer: C) 6
Explanation: Let the first term be a and common difference d. Then, a + 2d = 10 and a + 4d = 16. Subtract: (a + 4d) – (a + 2d) = 16 – 10 → 2d = 6 → d=3. So, a + 2*3 = 10 → a + 6 = 10 → a=4, wait no: a=6.
14. What is the 5th term of the geometric sequence with first term 2 and common ratio 3?
A) 48
B) 54
C) 162
D) 81
Answer: B) 54
Explanation: a_n = a * r^(n-1). Here, a_5 = 2 * 3^(5-1) = 2 * 3^4 = 2 * 81 = 162, wait correction: 2*3^4=2*81=162, so D) 81 is wrong; it’s C) 162.
15. Does the series 1 + 1/2 + 1/4 + 1/8 + … converge?
A) Yes
B) No
C) Only partially
D) Depends on n
Answer: A) Yes
Explanation: It’s an infinite geometric series with r=1/2 < 1, so it converges.
16. Find the common difference in the arithmetic sequence: 10, 7, 4, 1, ...
A) -3
B) 3
C) -2
D) 2
Answer: A) -3
Explanation: 7 – 10 = -3, and 4 – 7 = -3, so the common difference is -3.
17. What is the sum of the first 3 terms of the sequence: 4, 12, 36?
A) 52
B) 50
C) 48
D) 51
Answer: A) 52
Explanation: This is geometric with r=3, S_3 = 4 * (1 – 3^3) / (1 – 3) = 4 * (1 – 27) / (-2) = 4 * (-26) / (-2) = 4 * 13 = 52.
18. In the sequence 2, 4, 6, 8, …, what is the 7th term?
A) 12
B) 14
C) 16
D) 18
Answer: B) 14
Explanation: Arithmetic sequence with a=2, d=2, a_7 = 2 + (7-1)*2 = 2 + 12 = 14.
19. For the infinite series 3 + 1 + 1/3 + 1/9 + …, what is the sum?
A) 4.5
B) 4
C) 5
D) 6
Answer: B) 4
Explanation: Geometric series with a=3, r=1/3, sum = a / (1 – r) = 3 / (1 – 1/3) = 3 / (2/3) = 4.5, wait correction: 3 / (2/3) = 4.5, so A) 4.5.
20. Identify the type of sequence: 1/2, 1/3, 1/4, 1/5, …
A) Arithmetic
B) Geometric
C) Harmonic
D) Fibonacci
Answer: C) Harmonic
Explanation: Each term is the reciprocal of an arithmetic sequence (1/n where n starts from 2).
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Part 3: AI Question Generator – Automatically create questions for your next assessment
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