Class 11 Mathematics Sequences and Series MCQs Set 05

Question. The sum to infinite term of the series 1+2/3 +6/3² +10/3³ + 14/3⁴ + … is
(a) 3
(b) 4
(c) 6
(d) 2
Answer: a

Question. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then the common ratio is
(a) 5
(b) 1
(c) 4
(d) 3
Answer: c

Question. The sum of the series :
( b)² + 2(d)² + 3(6)² + … upto 10 terms is :
(a) 11300
(b) 11200
(c) 12100
(d) 12300
Answer: c

Question. Find the sum up to 16 terms of the series 1³/1 + 1³ + 2³/1+3 + 1³ + 2³ + 3³ /1+3+5+…
(a) 448
(b) 445
(c) 446
(d) None of these
Answer: c

Question. In a G.P. if (m + n)^th term is p and (m – n)^th term is q, then m^th term is:
(a) p/q
(b) q/p
(c) pq
(d) √pq
Answer: d

Question. If sum of the infinite G.P. is 4/3 and its first term is 3/4 then its common ratio is :
(a) 7/16
(b) 9/16
(c) 1/9
(d) 7/9
Answer: a

Question. If S₁,S₂ and S₃ denote the sum of first n₁, n₂ and n₃ terms respectively of an A.P., then value of s₁/n₁(n₂−n₃) +s₂/n₂(n₃−n₁) + s₃/n₃(n₁−n₂)is
(a) 1/2
(b) 0
(c) −1/2
(d) 3/2
Answer: b

Question. If the p^th, q^th and r^th terms of a G.P. are again in G.P., then which one of the following is correct?
(a) p, q, r are in A.P.
(b) p, q, r are in G.P.
(c) p, q, r are in H.P.
(d) p, q, r are neither in A.P. nor in G.P. nor in H.P.
Answer: a

Question. If the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, then the ratio of the first term to the common difference is :
(a) 1 : 2
(b) 2 : 1
(c) 1 : 4
Answer: a

Question. The series ( √2 +1),1,(√2 −1) … is in :
(a) A.P.
(b) G.P.
(c) H.P.
(d) None of these
Answer: b

Question. If sixth term of a H. P. is 1/61 and its tenth term is 1/105 then the first term of that H.P. is
(a) 1/28
(b) 1/39
(c) 1/6
(d) 1/17
Answer: c

Question. In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is:
(a) 7
(b) 21
(c) 28
(d) 42
Answer: c

Question. The coefficient of x⁵⁰ in the binomial expansion of (1 + x)¹⁰⁰⁰ + x (1 + x)⁹⁹⁹ + x²(1 + x)⁹⁹⁸ + …. + x¹⁰⁰⁰ is:
(a) (1000)! / (50)!(950)!
(b) (1000)! / (49)!(951)!
(c) (1000)! / (51)!(950)!
(d) (1000)! / (50)!(951)!
Answer: d

Question. 5^1+x + 5^1–x, a/2 , 5^2x + 5^–2x are in A.P., then the value of a is:
(a) a < 12
(b) a ≤ 12
(c) a ≥ 12
(d) None of these
Answer: d

Question. If in a series S_n = an² + bn + c, where S_n denotes the sum of n terms, then
(a) The series is always arithmetic
(b) The series is arithmetic from the second term onwards
(c) The series may or may not be arithmetic
(d) The series cannot be arithmetic
Answer: b

Question. What is the sum of the first 50 terms of the series (1 × 3) + (3 × 5) + (5 × 7) + …. ?
(a) 1,71,650
(b) 26,600
(c) 26,650
(d) 26,900
Answer: a

Question. The first term of an infinite G.P. is 1 and each term is twice the sum of the succeeding terms. then the sum of the series is
(a) 2
(b) 3
(c) 3/2
(d) 5/2
Answer: c

Question. The product of n positive numbers is unity, then their sum is :
(a) a positive integer
(b) divisible by n
(c) equal to n + 1/n
(d) never less than n
Answer: d

Question. The sum 1 + (1³+2³)/(1+2) + (1³+2³+3³)/(1+2+3) + …………… + (1³+2³+3³+….+15³)/(1+2+3+….+15 )− 1/2 (1+2+3+….+15) is equal to :
(a) 620
(b) 1240
(c) 1860
(d) 660
Answer: a

Question. The value of x + y + z is 15 if a, x, y, z, b are in A.P. while the value of 1/x+1/y+1/z is 5/3 if a, x, y, z, b are in H.P. Then the value of a and b are
(a) 2 and 8
(b) 1 and 9
(c) 3 and 7
(d) None of these
Answer: b

Question. The sum of 13 + 23 + 33+43 ... +153 is:
a. 22000
b. 10000
c. 14400
d. 15000

Answer: C

Question. If the A.M., G.M. and H.M. between two positive numbers a and b are equal, then:
a. a = b
b. ab = 1
c. a > b
d. a < b

Answer: A

Question. If the A.M. of two numbers is greater than G.M. of the numbers by 2 and the ratio of the numbers is 4 : 1, then the numbers are:
a. 4, 1
b. 12, 3
c. 16, 4
d. None of these

Answer: C

Question. If | x |<1, then the sum of the series 1+ 2x + 3x2 + 4x3 +...∞ will be:
a. 1/1–x
b. 1/1+x
c. 1/(1+x)²
d. 1/(1–x)²

Answer: D

Question. If the ratio of H.M. and G.M. of two quantities is 12 : 13,then the ratio of the numbers is:
a. 1: 2
b. 2: 3
c. 3: 4
d. None of these

Answer: D

Question. The pth term Tp of HP is q(p+q) and qth term Tq is. p(p+q) when p >1,q >1, then: A,B,
a. Tp+q + pq =
b. Tpq = p + q
c. Tp+q > Tpq
d. Tpq > Tp+q

Answer: C

Question. If the arithmetic, geometric and harmonic means between two positive real numbers be A, G and H , then:
a. 2 A = GH
b. 2 H = AG
c. G = AH
d. 2 G = AH

Answer: D

Question. If xloga ,x logb , xlogc , x be in H.P., then a, b, c are in:
a. A.P.
b. H.P.
c. G.P.
d. None of these

Answer: C

Question. If A1, A2 are the two A.M.'s between two numbers a and b and G1 ,G2 be two G.M.'s between same two numbers, then A1+A2/G1.G2 ?
a. a+b/ab
b. a+b/2ab
c. 2ab/a+b
d. ab/a+b

Answer: A

Question. If G.M. = 18 and A.M. = 27, then H.M. is:
a. 1/18
b. 1/12
c.12
d. 9√6

Answer: C

Question. Sum of n terms of series 12 + 16 + 24 + 40 + ..... will be:
a. 2 (2n − 1) + 8n
b. 2(2n − 1) + 6n
c. 3(2n − 1) + 8n
d. 4(2n − 1) + 8n

Answer: D

Question. The sum of the series 1+1.3/6 + 1.3.5/6.8 + ....∞ is
a. 1
b. 0
c. ∞
d. 4

Answer: D

Question. If the first and (2n –1)th the term of an AP, a GP and a HP are equal and their nth terms are a, b and c respectively, then:
a. a = b = c
b. a + c = b
c. a ≥ b ≥ c
d. ac = b2

Answer: D

Question. If the set of natural numbers is partitioned into subsets S1 ={1},S2 = {2, 3},S3 = {4, 5, 6} and so on. Then the sum of the terms in 50 S is:
a. 62525
b. 25625
c. 62500
d. None of these

Answer: A

Question. Sum of the squares of first n natural numbers exceeds their sum by 330, then n = ?
a.8
b.10
c.15
d.20

Answer: B

Question. If x = log 3+ log7 5 + log9 7 then:
a. x≥3/2
b. x≥1/3√2
c. x≥3/3√2
d. 1/√3

Answer: C

Question. First term of the 11th group in the following groups (1), (2, 3, 4), (5, 6, 7, 8, 9),……….is:
a.89
b. 97
c.101
d.123

Answer: C

Question. The third term of a geometric progression is 4. The product of the first five terms is :
a. 4³
b. 4⁵
c. 4⁴
d. 4⁷

Answer: B

Question. The H. M between roots of the equation x² – 10x + 11 = 0 is equal to :
a. 1/5
b. 5/21
c. 21/20
d. 11/5

Answer: D

Question. For a, b, c to be in G.P. What should be the value of a − b/b − c ?
a. ab
b. bc
c. a/b or b/c
d. None of these

Answer: C

Question. If a, b, c are in geometric progression and a, 2b, 3c are in arithmetic progression, then what is the common ratio r such that 0 < r < 1 ?
a. 1/3
b. 1/2
c. 1/4
d. 1/8

Answer: A

Question. In a Geometric Progression with first term a and common ratio r, what is the Arithmetic Mean of the first five terms?
a. a + 2r
b. a r²
c. a r⁵ – 1./[5r – 1.]
d. a r⁴ – 1./[5r – 1.]

Answer: C

Question. Which term of the following sequence 1/3 , 1/9 , 1/27 .......is 1/19683 ?
a. 3
b. 9
c. 6
d. None of these

Answer: B

Question. The value of 3 –1 + 1/3 – 1/9 + ........ + is equal to:
a. 20/9
b. 9/20
c. 9/4
d. 4/9

Answer: C

Question. The following consecutive terms 1/1+ √x, 1/1− x, 1/1− x of a series are in:
a. H.P.
b. G.P.
c. A.P.
d. A.P., G.P.

Answer: C

Question. How many terms of G.P. 3, 3², 3³, .......... are needed to give the sum 120?
a. 3
b. 4
c. 5
d. 6

Answer: B

STATEMENT TYPE QUESTIONS

Question. Statement I: If ‘a’ is the first term and ‘d’ is the common difference of an A.P., then its nth term is given by a_n = a – n – 1.d
Statement II: The sum S_n of n terms of an A.P. with first term ‘a’ and common difference ‘d’ is given by S_n = n/2 {2a + n – 1.d}
Choose the correct option.
a. Only I is true
b. Only II is true
c. Both are true
d. Both are false

Answer: B

Question. I. 11^th terms of the G.P. 5, 10, 20, 40, ... is 5120
II. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtained quadratic equation is x² – 16x + 25 = 0
Choose the correct option.
a. Only I is true
b. Only II is true.
c. Both are true
d. Both are false.

Answer: C

Question. Consider the following statements.
I. The nth term of a G.P. with first term ‘a’ and common ratio ‘r’ is given by a_n = a.r^n–1.
II. Geometric mean of a and b is given by ab.^1/3
Choose the correct option.
a. Only I is true
b. Only II is true
c. Both are true
d. Both are false

Answer: A

Question. Consider the following statements
I. If a₁, a₂, ..., a_n ... is a sequence, then the expression a₁ + a₂ + ... + a_n +… is called a series.
II. Those sequences whose terms follow certain patterns are called progressions.
Choose the correct option.
a. Only I is false
b. Only II is false
c. Both are false
d. Both are true

Answer: D

Question. I. If each term of a G.P. be raised to the same power, the resulting sequence also forms a G.P.
II. 25th term of the sequence 4, 9, 14, 19, ... is 124.
Choose the correct option.
a. Both are true
b. Both are false
c. Only I is true
d. Only II is true

Answer: A