ECAT Mathematics
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ECAT Mathematics
Chapter 6: Sequences and Series
100 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
A sequence is:
A
A set of numbers in no order
B
An ordered list of numbers
C
A random collection
D
A single number
1
In an Arithmetic Progression (AP), consecutive terms differ by:
A
A constant ratio
B
A constant difference
C
A variable amount
D
Zero
2
The common difference of AP 3, 7, 11, 15,... is:
A
3
B
4
C
7
D
11
3
The nth term of AP is:
A
a + nd
B
a + (n-1)d
C
a - (n-1)d
D
nd
4
The sum of first n terms of AP is:
A
n/2 × (a+l)
B
n × (a+l)
C
n/2 × (2a+(n-1)d)
D
Both A and C
5
In a Geometric Progression (GP), consecutive terms have:
A
Constant difference
B
Constant ratio (r)
C
Constant sum
D
Increasing values
6
The common ratio of GP 2, 6, 18, 54,... is:
A
2
B
3
C
4
D
6
7
The nth term of GP is:
A
a + (n-1)r
B
ar^n
C
ar^{n-1}
D
a^{n-1}r
8
The sum of first n terms of GP (r≠1) is:
A
a(r^n-1)/(r-1)
B
a(r^n+1)/(r+1)
C
n×ar
D
a(r^{n-1})/(r)
9
The sum to infinity of GP |r|<1 is:
A
a/(1-r)
B
a/(1+r)
C
a(1-r)
D
Infinite
10
For GP: 1, 1/2, 1/4,..., the sum to infinity is:
A
1
B
2
C
3
D
4
11
The Arithmetic Mean (AM) between a and b is:
A
(a+b)/2
B
ab
C
sqrt(ab)
D
2ab/(a+b)
12
The Geometric Mean (GM) between a and b (both positive) is:
A
(a+b)/2
B
sqrt(ab)
C
ab
D
(a-b)/2
13
The Harmonic Mean (HM) between a and b is:
A
(a+b)/2
B
sqrt(ab)
C
2ab/(a+b)
D
(a-b)/2
14
The relationship AM ≥ GM ≥ HM holds for:
A
All real numbers
B
Negative numbers only
C
Positive real numbers
D
Integers only
15
The 10th term of AP 2, 5, 8, 11,... is:
A
29
B
32
C
26
D
35
16
The sum of first 10 terms of AP 2, 5, 8,... is:
A
155
B
165
C
145
D
175
17
For GP 3, 6, 12, 24,..., the 5th term is:
A
48
B
36
C
96
D
72
18
For GP 3, 6, 12,..., sum of first 5 terms:
A
93
B
96
C
90
D
99
19
If 5th term of AP is 13 and 8th term is 22, common difference is:
A
2
B
3
C
4
D
5
20
The nth term of 1+3+5+... (odd numbers) is:
A
2n
B
2n-1
C
n^2
D
2n+1
21
Sum of first n natural numbers =
A
n(n+1)
B
n(n+1)/2
C
n^2
D
n(n-1)/2
22
Sum of first n odd numbers =
A
n^2
B
n(n+1)
C
n(n-1)/2
D
2n^2
23
Sum of squares of first n natural numbers:
A
n(n+1)/2
B
n(n+1)(2n+1)/6
C
n^2(n+1)^2/4
D
n(n+1)^2
24
Sum of cubes of first n natural numbers:
A
[n(n+1)/2]^2
B
n(n+1)(2n+1)/6
C
n^2(n+1)/2
D
n(n+1)/4
25
A series is:
A
A sequence only
B
The sum of terms of a sequence
C
A fraction
D
A geometric shape
26
The Harmonic Progression (HP) is related to AP by:
A
HP terms are reciprocals of an AP
B
HP is the same as AP
C
HP has constant differences
D
HP terms multiply
27
If 1/a, 1/b, 1/c are in AP, then a, b, c are in:
A
AP
B
GP
C
HP
D
Neither
28
In AP: a=3, d=4. The 20th term is:
A
79
B
83
C
75
D
80
29
Which is a GP?
A
1,2,3,4
B
2,4,6,8
C
1,2,4,8
D
1,3,5,7
30
Insert 3 arithmetic means between 2 and 18:
A
5,8,11
B
4,8,12
C
3,7,11
D
6,10,14
31
The middle term of AP with n=5 terms is:
A
2nd term
B
3rd term
C
4th term
D
5th term
32
Sum of n terms of GP: S_n = 3(2^n-1). The first term is:
A
1
B
3
C
6
D
2
33
Common ratio of GP where T_3=12 and T_1=3:
A
2
B
4
C
3
D
1/2
34
For AP with a=7 and l=63 with n terms, if S_n=350:
A
n=10
B
n=8
C
n=12
D
n=5
35
A geometric series 1+r+r^2+... converges when:
A
r<1
B
r>1
C
|r|<1
D
r=1
36
The sum 0.999... (repeating) equals:
A
0.999 approximately
B
1 exactly
C
Less than 1
D
1.001
37
In AP, if S_n=2n^2+3n, then T_n for n>1:
A
4n+1
B
4n-1
C
4n+3
D
2n+3
38
Insert 2 geometric means between 2 and 54:
A
6 and 18
B
3 and 9
C
4 and 12
D
8 and 24
39
For a GP with first term 1 and last term 128 with r=2, n is:
A
8
B
7
C
9
D
6
40
The sum of 0.3+0.03+0.003+... is:
A
1/3
B
1/30
C
0.33...=1/3
D
0.3
41
If a, b, c are in GP, then b^2 =
A
ac
B
a+c
C
2ac/(a+c)
D
a-c
42
If a, b, c are in AP, then 2b =
A
a+c
B
ac
C
a-c
D
a×c
43
If a, b, c are in HP, then b =
A
(a+c)/2
B
2ac/(a+c)
C
√(ac)
D
ac/(a+c)
44
The number of terms in AP: 3,6,9,...,99 is:
A
30
B
33
C
36
D
99
45
Sum of first 100 natural numbers:
A
5000
B
5050
C
4950
D
10100
46
The common difference of AP: -5,-1,3,7,... is:
A
4
B
5
C
-4
D
3
47
For GP: 2,-6,18,-54,..., common ratio r=
A
3
B
-3
C
4
D
-4
48
For |r|>1, the GP infinite series:
A
Converges to finite sum
B
Diverges (infinite sum)
C
Converges to zero
D
Oscillates finitely
49
The AM-GM inequality states AM ≥ GM with equality when:
A
a=b=0
B
a≠b
C
a=b
D
a>b
50
Sum of first n even numbers 2+4+6+...+2n:
A
n(n+1)
B
n^2
C
n(n-1)
D
2n
51
For the series 1+1/2+1/4+..., what is S_∞?
A
1
B
2
C
1/2
D
4
52
The nth term of 1×2+2×3+3×4+... is:
A
n(n+1)
B
n(n-1)
C
n^2
D
n^2+n+1
53
Sum of 1×2+2×3+...+n(n+1):
A
n(n+1)(n+2)/3
B
n(n+1)/2
C
n(n+1)(2n+1)/6
D
n^2(n+1)/2
54
If a+b=10 and ab=24, AM of a,b is:
A
5
B
24
C
4
D
10
55
If AM=5 and GM=4 for two numbers, HM =
A
16/5
B
5/16
C
20/5
D
5/4
56
The general term of the series 1+3+5+7+... is:
A
n
B
2n-1
C
2n
D
2n+1
57
How many terms of AP 17, 15, 13,... give sum 72?
A
9
B
8
C
7
D
10
58
Sequence 1, 4, 9, 16,... is:
A
AP
B
GP
C
Neither (squares of integers)
D
HP
59
For an AP, if T_5=23 and T_7=33, find T_3:
A
11
B
13
C
15
D
10
60
Sum of GP: a+ar+ar^2+...+ar^{n-1} when r=1 is:
A
na
B
a
C
nar
D
0
61
For decreasing GP: 12, 4, 4/3,..., S_∞ is:
A
18
B
16
C
24
D
12
62
A sequence where each term is the sum of two preceding is:
A
AP
B
GP
C
Fibonacci-type
D
HP
63
For AP: T_p=q and T_q=p, find T_{p+q}:
A
0
B
p+q
C
p-q
D
q-p
64
In GP, if the product of first 3 terms is 27 and they are in GP, the middle term is:
A
3
B
9
C
1
D
27
65
Sum of arithmetic series 1+3+5+...+(2n-1) =
A
n
B
n^2
C
n(n-1)
D
2n-1
66
For geometric sequence: if T_2=6 and T_5=48, find r:
A
2
B
3
C
4
D
6
67
If r=2 and T_2=6, find first term:
A
3
B
6
C
4
D
2
68
Sum of first 6 terms of GP 3,6,12,...:
A
189
B
186
C
192
D
183
69
Arithmetic series 2+5+8+...+302 has number of terms:
A
101
B
100
C
102
D
99
70
Sum of that AP 2+5+8+...+302:
A
15251
B
15352
C
15253
D
15000
71
An AP has a=5, d=3, n=20. Find l (last term):
A
62
B
60
C
64
D
58
72
Find sum S_{20} for AP 5,8,11,...:
A
650
B
655
C
640
D
670
73
In GP 5, 5r, 5r^2,... the 4th term is 40 when r=
A
2
B
4
C
8
D
3
74
The 4th term of GP with a=2 and r=3:
A
54
B
81
C
18
D
162
75
The sum of first n terms of the series whose general term is 3n-2:
A
n(3n-1)/2
B
n(3n+1)/2
C
3n^2
D
n(n-1)
76
If a = 3 and S = 9 for infinite GP, then r =
A
1/3
B
2/3
C
1/2
D
3/4
77
The sum of series 1-1+1-1+1... diverges because:
A
r=−1, |r|<1
B
r=−1, |r|=1
C
r=−1, |r|>1
D
r=0
78
For 3 numbers in AP with sum 15, the middle number is:
A
5
B
6
C
4
D
7
79
For 3 numbers in GP with product 216, the middle number is:
A
6
B
8
C
9
D
12
80
Sum of first 100 odd numbers:
A
10000
B
9999
C
5050
D
100
81
Sum formula for cube: 1^3+2^3+...+n^3:
A
n^2(n+1)^2/4
B
n(n+1)/2
C
n(n+1)(2n+1)/6
D
n^2(n+1)/2
82
If the 4th and 7th terms of an AP are 11 and 20 respectively, find the first term:
A
2
B
3
C
4
D
5
83
The sum of an AP: a, a+d, ..., a+(n-1)d can also be written as Sn=
A
n/2(a+l) where l=last term
B
n(a+d)
C
na
D
nd
84
For GP: 1, 3, 9, ..., find S_7:
A
1093
B
729
C
1024
D
2187
85
Three numbers in GP such that their sum is 21 and product is 216: first term if a/r, a, ar:
A
a=6; a/r and ar depend on r
B
a=1
C
a=3
D
a=9
86
In that GP, if r=2: terms are 3, 6, 12. Sum=?
A
21
B
18
C
24
D
15
87
Number of terms between 100 and 300 divisible by 7:
A
28
B
29
C
30
D
27
88
Their sum (multiples of 7 from 105 to 294):
A
5586
B
5628
C
5530
D
5600
89
For sum 1+2+4+8+...+2^9:
A
1023
B
1024
C
2046
D
512
90
If x, y, z are in AP, then 2y =
A
x+z
B
xz
C
x-z
D
x/z
91
If x, y, z are in GP, then y^2 =
A
x+z
B
xz
C
x-z
D
x/z
92
If x, y, z are in HP, then 1/x, 1/y, 1/z are in:
A
AP
B
GP
C
HP
D
None
93
The 10th term of series 0.6, 0.66, 0.666,...:
A
0.6(10)
B
0.6666666666
C
6/9
D
0.66...6 (10 sixes)
94
Sum of first n terms of series 1+11+111+...:
A
(10^{n+1}-9n-10)/81
B
n(n+1)/2
C
10^n
D
(10^n-1)/9
95
For which values of r does the infinite GP converge?
A
|r|<1
B
|r|≤1
C
|r|>1
D
r>0
96
Sum of all integers from 1 to 50:
A
1225
B
1275
C
1325
D
2550
97
The nth term of series 2,12,36,80,... (n^3+n^2):
A
n^2(n+1)
B
n^3+n^2
C
n(n+1)(n+1)/1
D
Both A and B
98
In AP: if S_5:S_10 = 1:3, then T_3:T_8 =
A
1:3
B
3:5
C
5:9
D
1:2
99
Sum of all two-digit numbers divisible by 3:
A
1665
B
1620
C
1683
D
1260
0
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