1
If f(x) = 6x − 2 and g(x) = -2x − 6, what is g(f(-4))?
Medium
Explanationf(-4) = -26, so g(f(-4)) = g(-26) = -2(-26) − 6 = 46.
2
A sequence is defined by a₁ = -11 and aₙ = aₙ₋₁ − 10 for n > 1. What is a12?
Medium
ExplanationEach term adds -10 to the previous one, so a12 = a₁ + (12−1)(-10) = -121.
3
In an arithmetic sequence, the 6th term is -5 and the 17th term is -27. What is the 2nd term?
Medium
Explanationd = (-27 − (-5))/(17−6) = -2. Then the 2nd term = -5 + (2−6)(-2) = 3.
4
In an arithmetic sequence, the 7th term is -45 and the 22nd term is -225. What is the 17th term?
Medium
Explanationd = (-225 − (-45))/(22−7) = -12. Then the 17th term = -45 + (17−7)(-12) = -165.
5
What is the sum of the first 8 terms of a geometric sequence with first term 7 and common ratio 3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 7((3)^8 − 1)/(3 − 1) = 22960.
6
If f(x) = -x + 20, what is f(13) + f(-12)?
Medium
Explanationf(13) = 7 and f(-12) = 32, so f(13) + f(-12) = 39.
7
What is the sum of the first 8 terms of a geometric sequence with first term -1 and common ratio 3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = -1((3)^8 − 1)/(3 − 1) = -3280.
8
What is the sum of the first 13 terms of an arithmetic sequence with first term 24 and common difference 4?
Medium
ExplanationThe 13th term is 72. Sum = n(a₁ + aₙ)/2 = 13(24 + 72)/2 = 624.
9
What is the sum of the first 3 terms of a geometric sequence with first term 8 and common ratio 4?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 8((4)^3 − 1)/(4 − 1) = 168.
10
If f(x) = -4x + 5 and g(x) = f(x + 9) − 6, what is g(-15)?
Medium
Explanationg(-15) = f(-6) − 6 = 29 − 6 = 23.
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11
If f(x) = -8x − 13 and g(x) = 7x + 4, what is g(f(8))?
Medium
Explanationf(8) = -77, so g(f(8)) = g(-77) = 7(-77) + 4 = -535.
12
If f(x) = -x + 13 and g(x) = -7x + 11, what is f(g(5))?
Medium
Explanationg(5) = -24, so f(g(5)) = f(-24) = -1(-24) + 13 = 37.
13
If f(x) = -12x + 13, what is f(-4) + f(5)?
Medium
Explanationf(-4) = 61 and f(5) = -47, so f(-4) + f(5) = 14.
14
Is the sequence -7, -17, -11, -9, ... arithmetic, geometric, or neither?
Medium
ExplanationThere is no constant common difference or common ratio between terms, so this sequence is neither arithmetic nor geometric.
15
A geometric sequence has first term -3 and common ratio 4. What is the 8th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = -3 × (4)^7 = -49152.
16
A sequence is defined by a₁ = 7 and aₙ = aₙ₋₁ − 5 for n > 1. What is a4?
Medium
ExplanationEach term adds -5 to the previous one, so a4 = a₁ + (4−1)(-5) = -8.
17
If f(x) = -x^2 + 2x + 18, what is f(-6)?
Medium
Explanationf(-6) = -1(-6)^2 + (2)(-6) + (18) = -30.
18
If f(x) = -x + 4, what is f(-11)?
Medium
Explanationf(-11) = (-1)(-11) + 4 = 15.
19
In an arithmetic sequence, the 5th term is 33 and the 12th term is 61. What is the 11th term?
Medium
Explanationd = (61 − 33)/(12−5) = 4. Then the 11th term = 33 + (11−5)(4) = 57.
20
Is the sequence -10, -8, -6, -4, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term differs from the previous by a constant amount, so this sequence is arithmetic.
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21
What is the sum of the first 25 terms of an arithmetic sequence with first term -21 and common difference -10?
Medium
ExplanationThe 25th term is -261. Sum = n(a₁ + aₙ)/2 = 25(-21 + (-261))/2 = -3525.
22
An employee's salary starts at Rs. 47808 and increases by Rs. 3701 each year. What will the salary be after 8 years?
Medium
ExplanationSalary after 8 years = 47808 + 8 × 3701 = Rs. 77416.
23
For the function f(x) = 1/(x − 62), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − 62 cannot equal 0, so x = 62 must be excluded from the domain.
24
If f(x) = 7x + 12, what is f⁻¹(103)?
Medium
Explanationf⁻¹(103) is the x-value such that f(x) = 103: 7x + 12 = 103, so x = 13.
25
Is the sequence -4, 2, -4, -13, ... arithmetic, geometric, or neither?
Medium
ExplanationThere is no constant common difference or common ratio between terms, so this sequence is neither arithmetic nor geometric.
26
For the function f(x) = √(x − 9), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − 9 ≥ 0, so x ≥ 9. The smallest such value is 9.
27
If f(x) = 3x − 10 and g(x) = -6x + 11, what is f(g(-1))?
Medium
Explanationg(-1) = 17, so f(g(-1)) = f(17) = 3(17) − 10 = 41.
28
If f(x) = -6x − 8 and g(x) = f(x + 1) − 15, what is g(15)?
Medium
Explanationg(15) = f(16) − 15 = -104 − 15 = -119.
29
A population of 2500 grows by 20% each year. What will the population be after 4 years?
Medium
ExplanationPopulation after 4 years = 2500 × (1 + 20/100)^4 = 5184.
30
If f(x) = -x + 13, what is f(-8) + f(-2)?
Medium
Explanationf(-8) = 21 and f(-2) = 15, so f(-8) + f(-2) = 36.
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31
f(x) = 8x + 5 if x < 8, and f(x) = 6x + 7 if x ≥ 8. What is f(18)?
Medium
ExplanationSince 18 ≥ 8, use f(x) = 6x + 7: f(18) = 115.
32
In an arithmetic sequence, the 5th term is 8 and the 20th term is 38. What is the 14th term?
Medium
Explanationd = (38 − 8)/(20−5) = 2. Then the 14th term = 8 + (14−5)(2) = 26.
33
If f(x) = -2x − 25, what is f(-2)?
Medium
Explanationf(-2) = (-2)(-2) − 25 = -21.
34
An arithmetic sequence has first term 19 and common difference -13. What is the 18th term?
Medium
Explanationaₙ = a₁ + (n−1)d = 19 + (18−1)(-13) = -202.
35
A sequence is defined by a₁ = 14 and aₙ = aₙ₋₁ − 7 for n > 1. What is a14?
Medium
ExplanationEach term adds -7 to the previous one, so a14 = a₁ + (14−1)(-7) = -77.
36
If f(x) = -10x − 5, what is f(-15) − f(8)?
Medium
Explanationf(-15) = 145 and f(8) = -85, so f(-15) − f(8) = 230.
37
In an arithmetic sequence, the 4th term is -41 and the 9th term is -91. What is the 16th term?
Medium
Explanationd = (-91 − (-41))/(9−4) = -10. Then the 16th term = -41 + (16−4)(-10) = -161.
38
What is the sum of the first 5 terms of a geometric sequence with first term 8 and common ratio -3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 8((-3)^5 − 1)/(-3 − 1) = 488.
39
f(x) = -2x − 2 if x < -6, and f(x) = -5x + 1 if x ≥ -6. What is f(-18)?
Medium
ExplanationSince -18 < -6, use f(x) = -2x − 2: f(-18) = 34.
40
If f(x) = -5x + 2, what is f(6) + f(-4)?
Medium
Explanationf(6) = -28 and f(-4) = 22, so f(6) + f(-4) = -6.
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41
What is the sum of the first 12 terms of an arithmetic sequence with first term -3 and common difference 4?
Medium
ExplanationThe 12th term is 41. Sum = n(a₁ + aₙ)/2 = 12(-3 + 41)/2 = 228.
42
An employee's salary starts at Rs. 58315 and increases by Rs. 3780 each year. What will the salary be after 14 years?
Medium
ExplanationSalary after 14 years = 58315 + 14 × 3780 = Rs. 111235.
43
If f(x) = x − 23, for what value of x does f(x) = -26?
Medium
Explanationx − 23 = -26, so x = -3.
44
An employee's salary starts at Rs. 64052 and increases by Rs. 2780 each year. What will the salary be after 8 years?
Medium
ExplanationSalary after 8 years = 64052 + 8 × 2780 = Rs. 86292.
45
What is the sum of the first 8 terms of an arithmetic sequence with first term -23 and common difference -6?
Medium
ExplanationThe 8th term is -65. Sum = n(a₁ + aₙ)/2 = 8(-23 + (-65))/2 = -352.
46
If f(x) = 9x − 4, what is f⁻¹(86)?
Medium
Explanationf⁻¹(86) is the x-value such that f(x) = 86: 9x − 4 = 86, so x = 10.
47
A population of 900 grows by 20% each year. What will the population be after 2 years?
Medium
ExplanationPopulation after 2 years = 900 × (1 + 20/100)^2 = 1296.
48
What is the sum to infinity of a geometric sequence with first term 6 and common ratio -3/4?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 6/(1 − (-3)/4) = 24/7.
49
In an arithmetic sequence, the 5th term is 9 and the 18th term is 22. What is the 7th term?
Medium
Explanationd = (22 − 9)/(18−5) = 1. Then the 7th term = 9 + (7−5)(1) = 11.
50
If f(x) = 7x − 7 and g(x) = f(x + 11) + 14, what is g(13)?
Medium
Explanationg(13) = f(24) + 14 = 161 + 14 = 175.
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51
If f(x) = 8x − 8 and g(x) = f(x + 9) + 10, what is g(-14)?
Medium
Explanationg(-14) = f(-5) + 10 = -48 + 10 = -38.
52
An arithmetic sequence has first term 43 and common difference -2. What is the 29th term?
Medium
Explanationaₙ = a₁ + (n−1)d = 43 + (29−1)(-2) = -13.
53
What is the sum to infinity of a geometric sequence with first term 18 and common ratio 2/7?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 18/(1 − 2/7) = 126/5.
54
An arithmetic sequence has first term 29 and common difference 2. What is the 38th term?
Medium
Explanationaₙ = a₁ + (n−1)d = 29 + (38−1)(2) = 103.
55
Is the sequence -14, -18, -22, -26, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term differs from the previous by a constant amount, so this sequence is arithmetic.
56
A geometric sequence has first term 10 and second term 4. What is the common ratio?
Medium
ExplanationCommon ratio = a₂/a₁ = 4/10 = 2/5.
57
An arithmetic sequence has first term 40 and common difference -8. What is the 10th term?
Medium
Explanationaₙ = a₁ + (n−1)d = 40 + (10−1)(-8) = -32.
58
For the function f(x) = √(x − 8), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − 8 ≥ 0, so x ≥ 8. The smallest such value is 8.
59
An employee's salary starts at Rs. 20528 and increases by Rs. 1105 each year. What will the salary be after 6 years?
Medium
ExplanationSalary after 6 years = 20528 + 6 × 1105 = Rs. 27158.
60
If f(x) = -5x + 13 and g(x) = -5x + 15, what is g(f(6))?
Medium
Explanationf(6) = -17, so g(f(6)) = g(-17) = -5(-17) + 15 = 100.
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61
If f(x) = 6x + 5 and g(x) = f(x + 10) + 6, what is g(-13)?
Medium
Explanationg(-13) = f(-3) + 6 = -13 + 6 = -7.
62
What is the sum of the first 19 terms of an arithmetic sequence with first term -30 and common difference -10?
Medium
ExplanationThe 19th term is -210. Sum = n(a₁ + aₙ)/2 = 19(-30 + (-210))/2 = -2280.
63
An arithmetic sequence has first term -13 and common difference 11. What is the 17th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -13 + (17−1)(11) = 163.
64
An employee's salary starts at Rs. 52890 and increases by Rs. 624 each year. What will the salary be after 9 years?
Medium
ExplanationSalary after 9 years = 52890 + 9 × 624 = Rs. 58506.
65
If f(x) = 12x − 4, what is f⁻¹(-172)?
Medium
Explanationf⁻¹(-172) is the x-value such that f(x) = -172: 12x − 4 = -172, so x = -14.
66
An arithmetic sequence has first term 49 and common difference 10. What is the 16th term?
Medium
Explanationaₙ = a₁ + (n−1)d = 49 + (16−1)(10) = 199.
67
Is the sequence -15, -23, -31, -39, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term differs from the previous by a constant amount, so this sequence is arithmetic.
68
An arithmetic sequence has first term -40 and common difference 10. What is the 13th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -40 + (13−1)(10) = 80.
69
If f(x) = -4x − 18, what is f(15) + f(-14)?
Medium
Explanationf(15) = -78 and f(-14) = 38, so f(15) + f(-14) = -40.
70
If f(x) = -9x − 10, what is f(12) + f(-10)?
Medium
Explanationf(12) = -118 and f(-10) = 80, so f(12) + f(-10) = -38.
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71
If f(x) = -3x^2 − 6x, what is f(-9)?
Medium
Explanationf(-9) = -3(-9)^2 + (-6)(-9) + (0) = -189.
72
If f(x) = 4x and g(x) = f(x − 12) − 4, what is g(-11)?
Medium
Explanationg(-11) = f(-23) − 4 = -92 − 4 = -96.
73
f(x) = -6x + 2 if x < 2, and f(x) = -3x − 10 if x ≥ 2. What is f(1)?
Medium
ExplanationSince 1 < 2, use f(x) = -6x + 2: f(1) = -4.
74
In an arithmetic sequence, the 13th term is -42 and the 17th term is -70. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-70 − (-42))/(17−13) = -7.
75
An employee's salary starts at Rs. 68723 and increases by Rs. 1848 each year. What will the salary be after 6 years?
Medium
ExplanationSalary after 6 years = 68723 + 6 × 1848 = Rs. 79811.
76
An arithmetic sequence has first term -13 and common difference 3. What is the 19th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -13 + (19−1)(3) = 41.
77
If f(x) = -2x − 12 and g(x) = x, what is f(g(-9))?
Medium
Explanationg(-9) = -9, so f(g(-9)) = f(-9) = -2(-9) − 12 = 6.
78
In an arithmetic sequence, the 3rd term is 8 and the 13th term is -62. What is the 26th term?
Medium
Explanationd = (-62 − 8)/(13−3) = -7. Then the 26th term = 8 + (26−3)(-7) = -153.
79
An employee's salary starts at Rs. 74506 and increases by Rs. 1030 each year. What will the salary be after 4 years?
Medium
ExplanationSalary after 4 years = 74506 + 4 × 1030 = Rs. 78626.
80
An employee's salary starts at Rs. 50067 and increases by Rs. 2277 each year. What will the salary be after 14 years?
Medium
ExplanationSalary after 14 years = 50067 + 14 × 2277 = Rs. 81945.
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81
If f(x) = -3x − 9 and g(x) = 6x + 7, what is g(f(-3))?
Medium
Explanationf(-3) = 0, so g(f(-3)) = g(0) = 6(0) + 7 = 7.
82
An employee's salary starts at Rs. 31127 and increases by Rs. 1974 each year. What will the salary be after 4 years?
Medium
ExplanationSalary after 4 years = 31127 + 4 × 1974 = Rs. 39023.
83
An arithmetic sequence has first term 40 and common difference 8. What is the 29th term?
Medium
Explanationaₙ = a₁ + (n−1)d = 40 + (29−1)(8) = 264.
84
What is the sum of the first 7 terms of a geometric sequence with first term 2 and common ratio 3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 2((3)^7 − 1)/(3 − 1) = 2186.
85
What is the sum of the first 8 terms of an arithmetic sequence with first term 16 and common difference 4?
Medium
ExplanationThe 8th term is 44. Sum = n(a₁ + aₙ)/2 = 8(16 + 44)/2 = 240.
86
What is the sum of the first 17 terms of an arithmetic sequence with first term -1 and common difference 5?
Medium
ExplanationThe 17th term is 79. Sum = n(a₁ + aₙ)/2 = 17(-1 + 79)/2 = 663.
87
A geometric sequence has first term 3 and second term 12. What is the common ratio?
Medium
ExplanationCommon ratio = a₂/a₁ = 12/3 = 4.
88
For the function f(x) = √(x − (-10)), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − (-10) ≥ 0, so x ≥ -10. The smallest such value is -10.
89
A population of 21000 grows by 10% each year. What will the population be after 3 years?
Medium
ExplanationPopulation after 3 years = 21000 × (1 + 10/100)^3 = 27951.
90
If f(x) = 3x − 1, what is f⁻¹(38)?
Medium
Explanationf⁻¹(38) is the x-value such that f(x) = 38: 3x − 1 = 38, so x = 13.
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91
In an arithmetic sequence, the 12th term is -40 and the 22nd term is -110. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-110 − (-40))/(22−12) = -7.
92
What is the sum to infinity of a geometric sequence with first term 13 and common ratio -3/5?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 13/(1 − (-3)/5) = 65/8.
93
If f(x) = -8x − 6 and g(x) = -3x + 3, what is g(f(-9))?
Medium
Explanationf(-9) = 66, so g(f(-9)) = g(66) = -3(66) + 3 = -195.
94
Is the sequence -7, 21, -63, 189, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term is a constant multiple of the previous one, so this sequence is geometric.
95
If f(x) = 3x + 15 and g(x) = f(x + 12) − 3, what is g(0)?
Medium
Explanationg(0) = f(12) − 3 = 51 − 3 = 48.
96
A geometric sequence has first term -9 and second term 27. What is the common ratio?
Medium
ExplanationCommon ratio = a₂/a₁ = 27/-9 = -3.
97
An arithmetic sequence has first term -29 and common difference -10. What is the 20th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -29 + (20−1)(-10) = -219.
98
If f(x) = 5x + 22, for what value of x does f(x) = 112?
Medium
Explanation5x + 22 = 112, so x = 18.
99
If f(x) = 11x − 17, what is f(10) − f(12)?
Medium
Explanationf(10) = 93 and f(12) = 115, so f(10) − f(12) = -22.
100
What is the sum of the first 7 terms of an arithmetic sequence with first term -23 and common difference -2?
Medium
ExplanationThe 7th term is -35. Sum = n(a₁ + aₙ)/2 = 7(-23 + (-35))/2 = -203.