250+ TOP MCQs on Harmonic Sequences and Answers

Discrete Mathematics Multiple Choice Questions on “Harmonic Sequences”.

1. If a₁, a₂……… are in AP then a₁^-1, a₂^-1……… are in __________
a) An airthmetic sequence
b) A geometic progression
c) Airthmetico-geometric progression
d) None of the mentioned

Answer: d
Clarification: If a₁, a₂……… are in AP, then a₁^-1, a₂^-1……… are in Harmonic Progression.

2. The ninth term of ¹⁄₃, ¹⁄₇, ¹⁄₁₁, ¹⁄₁₅, ¹⁄₁₉,……… is given by?
a) ¹⁄₃₅
b) ¹⁄₃₆
c) ¹⁄₃₉
d) None of the mentioned

Answer: a
Clarification: Since here a₁^-1, a₂^-1……… are in AP thus a₉ = 3 + (9-1)4 = 35, ¹⁄₃₅ is h₉ term of the series.

3. If for some number a and d, if first term is ¹⁄_a, second term is 1/(a+d), thrid term is 1/(a+2d) and so on, then 5^th term of the sequence is?
a) a+4d
b) a-4d
c) 1/(a+4d)
d) none of the mentioned

Answer: c
Clarification: The given sequence will form HP, thus 5^th term will be (a+(5-1)d) – 1.

4. If a, b, c are in hp then a^-1, b^-1, c^-1 are in _________
a) GP
b) HP
c) AP
d) None of the mentioned

Answer: c
Clarification: If a₁, a₂……… are in AP then a₁^-1, a₂^-1……… are in Harmonic Progression.

5. If a, b, c are in hp, then b is related with a and c as _________
a) 2(¹⁄_b) = (¹⁄_a + ¹⁄_c)
b) 2(¹⁄_c) = (¹⁄_b + ¹⁄_c)
c) 2(¹⁄_a) = (¹⁄_a + ¹⁄_b)
d) None of the mentioned

Answer: a
Clarification: ¹⁄_a, ¹⁄_b, ¹⁄_c willl be in airthmentic series and ¹⁄_b will be the AM of a, c.

6. For number A, C if H is harmonic mean, G is geometric mean then H>=G.
a) True
b) False

Answer: b
Clarification: Geometric mean is always greater than or equal to the harmonic mean.

7. For number B, C if H is harmonic mean, A is the airthmetic mean then H>=A.
a) True
b) False

Answer: b
Clarification: Airthmetic mean is always greater than or equal to harmonic mean.

8. Which of the following gives the right inequality for AM, GM, HM?
a) AM>=HM>=GM
b) GM>=AM>=HM
c) AM>=GM>=HM
d) GM>=HM>=AM

Answer: c
Clarification: Airthmetic mean is always greater than or equal to geometric mean,geometric mean is always greater than or equal to harmonic mean.

9. For two number a,b HM between them is given by?
a) (2b+2a )/3b
b) 2ab/(a+b)
c) (a+b)/2ab
d) 2b/(a+b)

Answer: b
Clarification: Let c be the hm, ²⁄_c = ¹⁄_a + ¹⁄_b (AM property), c = 2b/(a+b).

10. If A, G, H are the AM, GM, HM between a and b respectively then?
a) A, G, H are in hp
b) A, G, H are in gp
c) A, G, H are in ap
d) None of the mentioned

Answer: b
Clarification: A = (a+b)/2, G = (ab)^1/2, H = 2b/(a+b), clearly AxH = G² thus A, G, H are in gp.