Sequences and Series Practice Questions
ATAR (Australia) · ATAR Mathematics Methods · 142 free MCQs with instant results and detailed explanations.
142
Total
45
Easy
75
Medium
22
Hard
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Sample Questions from Sequences and Series
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Q1
Easy
What is the 5th term of the arithmetic sequence where the first term is 3 and the common difference is 2?
A.
9
B.
10
C.
11
D.
12
Show Answer & Explanation
Correct Answer: C
In an arithmetic sequence, the nth term can be found using the formula: a_n = a_1 + (n-1)d. Here, a_1 = 3, d = 2, and n = 5. Thus, a_5 = 3 + (5-1) * 2 = 3 + 8 = 11.
Q2
Easy
In a geometric sequence, if the first term is 4 and the common ratio is 3, what is the 5th term?
A.
324
B.
243
C.
108
D.
81
Show Answer & Explanation
Correct Answer: A
The nth term of a geometric sequence is calculated using a_n = a_1 * r^(n-1). Here, a_1 = 4, r = 3, and n = 5. Thus, a_5 = 4 * 3^(5-1) = 4 * 81 = 324.
Q3
Easy
If the first five terms of a geometric sequence are 2, 6, 18, 54, and x, what is the value of x?
A.
162
B.
80
C.
128
D.
144
Show Answer & Explanation
Correct Answer: A
In a geometric sequence, each term is found by multiplying the previous term by the common ratio. The common ratio (r) can be found: 6/2 = 3. Therefore, x = 54 * 3 = 162.
Q4
Medium
What is the 10th term of the arithmetic sequence where the first term is 5 and the common difference is 3?
A.
32
B.
27
C.
35
D.
20
Show Answer & Explanation
Correct Answer: A
To find the nth term of an arithmetic sequence, use the formula: T_n = a + (n-1)d. Here, a = 5, d = 3, and n = 10. Thus, T_10 = 5 + (10-1)3 = 5 + 27 = 32.
Q5
Medium
If the sum of the first n terms of a geometric series is given by S_n = 3(2^n - 1), what is the common ratio of the series?
A.
2
B.
3
C.
1.5
D.
4
Show Answer & Explanation
Correct Answer: A
The formula for the sum of a geometric series S_n = a(1 - r^n) / (1 - r) can be rearranged. Here, the formula shows that the common ratio (r) is 2 since it must match the growth factor in 2^n.
Q6
Medium
A quadratic sequence has the first three terms as 2, 5, and 10. What is the 4th term of the sequence?
A.
17
B.
18
C.
20
D.
21
Show Answer & Explanation
Correct Answer: A
To find the 4th term, we observe the differences: 5-2=3, 10-5=5. The second differences (5-3=2) are constant, indicating a quadratic sequence. The next first difference would be 7 (5+2), leading to the 4th term being 10+7=17.
Q7
Medium
In a sequence defined by a recursive relation, T_n = 2T_(n-1) + 1 with T_1 = 1, what is T_4?
A.
15
B.
13
C.
14
D.
16
Show Answer & Explanation
Correct Answer: A
Using the recursive definition, calculate T_2 = 2(1) + 1 = 3, T_3 = 2(3) + 1 = 7, and T_4 = 2(7) + 1 = 15. Therefore, T_4 = 15.
Q8
Hard
In an arithmetic sequence, the first term is 5 and the common difference is 3. What is the 20th term of this sequence?
A.
62
B.
59
C.
65
D.
68
Show Answer & Explanation
Correct Answer: A
The nth term of an arithmetic sequence can be calculated using the formula: a_n = a_1 + (n-1)d. Here, a_1 = 5, d = 3, and n = 20. Therefore, a_20 = 5 + (20-1) * 3 = 5 + 57 = 62. Hence, the correct answer is A.
Q9
Hard
A geometric series has a first term of 4 and a common ratio of 2. What is the sum of the first 10 terms of this series?
A.
4092
B.
2044
C.
1020
D.
2048
Show Answer & Explanation
Correct Answer: A
The sum S_n of the first n terms of a geometric series can be calculated by S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term and r is the common ratio. Here, a_1 = 4, r = 2, and n = 10. Therefore, S_10 = 4 * (1 - 2^10) / (1 - 2) = 4 * (1 - 1024) / (-1) = 4 * 1023 = 4092. Thus, the correct answer is A.
Q10
Hard
Consider the arithmetic sequence where the first term is 5 and the common difference is 3. What is the 20th term of this sequence?
A.
62
B.
59
C.
68
D.
65
Show Answer & Explanation
Correct Answer: A
In an arithmetic sequence, the nth term can be calculated using the formula: a_n = a + (n-1)d. Here, a = 5, d = 3, and n = 20. Plugging in these values gives: a_20 = 5 + (20-1) * 3 = 5 + 57 = 62.
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Sequences and Series — ATAR (Australia) ATAR Mathematics Methods Practice Questions Online
This page contains 142 practice MCQs for the chapter Sequences and Series in ATAR (Australia) ATAR Mathematics Methods. The questions are organized by difficulty — 45 easy, 75 medium, 22 hard — so you can choose the right level for your preparation.
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