Discrete Mathematics Multiple Choice Questions on “Sequences and Summations”.
1. For the sequence 1, 7, 25, 79, 241, 727 … simple formula for {an} is ____________
a) 3n+1 – 2
b) 3n – 2
c) (-3)n + 4
d) n2 – 2
Answer: b
Clarification: The ratio of consecutive numbers is close to 3. Comparing these terms with the sequence of {3n} which is 3, 9, 27 …. Comparing these terms with the corresponding terms of sequence {3n} and the nth term is 2 less than the corresponding power of 3.
2. For the sequence 0, 1, 2, 3 an is ____________
a) ⌈n/2⌉+⌊n/2⌋
b) ⌈n/2⌉+⌈n/2⌉
c) ⌊n/2⌋+⌊n/2⌋
d) ⌊n/2⌋
Answer: a
Clarification: Expand the sequence ⌈n/2⌉+⌊n/2⌋ where a1 is ⌊0.5⌋+⌈0.5⌉ = 1+0 = 1, a2 is ⌊1⌋+⌈1⌉ = 1 + 1 = 2 and so on.
3. The value of∑(k=50)100 k2 is __________
a) 338, 350
b) 297, 900
c) 297, 925
d) 290, 025
Answer: c
Clarification: Using the formula. ∑(k=1)n k2 = (n(n + 1)(2n + 1)) / 6.
4. The sets A and B have same cardinality if and only if there is ___________ from A to B.
a) One-to-one
b) One-to-many
c) Many-to-many
d) Many-to-one
Answer: a
Clarification: If there is one-to-one correspondence then they have same cardinality.
5. For the sequence an = ⌊√2n+ 1/2⌋, a7is ____________
a) 1
b) 7
c) 5
d) 4
Answer: d
Clarification: a7 = ⌊√14+1/2⌋ which is ⌊4.24⌋ = 4.
6. The value of ∑(i=1)3 ∑(h=0)2 i is _________
a) 10
b) 17
c) 15
d) 18
Answer: d
Clarification: The value of ∑(i=1)3 ∑(h=0)2 i = 1+1+1+2+2+2+3+3+3 = 18.
7. For the sequence an = 6. (1/3)n, a4 is _________
a) 2/25
b) 2/27
c) 2/19
d) 2/13
Answer: b
Clarification: Put n = 4 in the sequence.
8. The value of ∑(i=0)4i! is __________
a) 32
b) 30
c) 34
d) 35
Answer: c
Clarification: First five term of the sequence n! is given by 1, 1, 2, 6, 24.
9. Set of all integers is counter.
a) True
b) False
Answer: a
Clarification: There is one-to-one correspondence between set of positive integers and set of all integers.
10. The value of ∏(k=1)100(-1) k is _________
a) 0
b) 1
c) -1
d) 2
Answer: b
Clarification: The product of a1, a2, a3 …… an is represented by ∏(i=1)n ai.