VBQs Sequences and Series Class 11 Mathematics

VBQs Sequences and Series Class 11 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 11 Mathematics with solutions. The following Sequences and Series Class 11 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 11 examinations.

Sequences and Series VBQs Class 11 Mathematics

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

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

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

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

Question. The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is
(a) –4
(b) –12
(c) 12
(d) 4

Question. What is the sum of terms equidistant from the beginning and end in an A.P. ?
(a) First term – Last term
(b) First term × Last term
(c) First term + Last term
(d) First term ÷ Last term

Question. Sum of n terms of series 1.3+3.5+5.7+………. is
(a) 1/3 n(n + 1)(2n + 1) − n
(b) 3/2 n(n + 1)(2n + 1) − n
(c) 4/5 n(n + 1)(2n + 1) − n
(d) 2/3 n(n + 1)(2n + 1) − n

Question. If the arithmetic mean of two numbers be A and geometric mean be G, then the numbers will be

Question. If 5 (3^{a – 1} + 1), (6^{2a – 3} +2) and 7(5^{a – 2} + 5) are in AP, then what is the value of a?
(a) 7
(b) 6
(c) 5
(d) None of these

Question. The A. M. between two positive numbers a and b is twice the G. M. between them. The ratio of the numbers is
(a) (√2 + 3) : (√2 – 3)
(b) (2 + √3 ) : (2 – √3 )
(c) (√3 + 1) : (√3 – 1)
(d) None of these

Question. There are four arithmetic means between 2 and –18. The means are
(a) –4, –7, –10, –13
(b) 1, –4, –7, –10
(c) –2, –5, –9, –13
(d) –2, –6, –10, –14

Question. If the ^jth term and kth term of an A.P. are k and j respectively, the ( k + j) th term is
(a) 0
(b) 1
(c) k + j + 1
(d) k + j –1

Question. The harmonic mean of a/1 – ab and a/1 + ab is

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

Question. Let x be one A.M and g₁ and g₂ be two G.Ms between y and z. What is g₁³ + g₂³ equal to ?
(a) xyz
(b) xy²z
(c) xyz²
(d) 2 xyz

Question. If 1, x, y, z, 16 are in geometric progression, then what is the value of x + y + z ?
(a) 8
(b) 12
(c) 14
(d) 16

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

Question. If the sequence is defined by an = n(n + 2), then match the columns. 

Codes
A B C D E
(a) 4 3 5 2 1
(b) 4 2 5 3 1
(c) 1 3 2 5 4
(d) 3 4 5 1 2

Question. If the sum of a certain number of terms of the A.P. 25, 22, 19, …….. is 116. then the last term is
(a) 0
(b) 2
(c) 4
(d) 6

Question. The product of first nine terms of a GP is, in general, equal to which one of the following?
(a) The 9th power of the 4th term
(b) The 4th power of the 9th term
(c) The 5th power of the 9th term
(d) The 9th power of the 5th term

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

Question. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4 ,then :
(a) a = 4/7, r = 3/7
(b) a = 2, r = 3/8 
(c) a = 3/2, r = 1/2 
(d) a = 3, r = 1/4

Question. If the sum of the first 2n terms of 2, 5, 8, ……. is equal to the sum of the first n terms of 57, 59, 61……., then n is equal to 
(a) 10
(b) 12
(c) 11
(d) 13

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

Question. The sum of the first n terms of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² +… is n(n+1)²/2 when n is even. When n is odd the sum is

Question. If the 7^th term of a H.P. is 1/10 and the 12^th term is 1/25, then the 20^th term is
(a) 1/37
(b) 1/41
(c) 1/45
(d) 1/49

Question. The first and eight terms of a G.P. are x^–4 and x⁵² respectively. If the second term is x^t, then t is equal to:
(a) –13
(b) 4
(c) 5/2
(d) 3

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

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

Question. If the nth term of an arithmetic progression is 3n + 7, then what is the sum of its first 50 terms?
(a) 3925
(b) 4100
(c) 4175
(d) 8200

Question. Let a, b, c, be in A.P. with a common difference d. Then e^{1/ c} , e^{b / ac} , e^{1/ a} are in :
(a) G.P. with common ratio e^d
(b) G.P with common ratio e^1/d
(c) G.P. with common ratio e^{d /(b2 − d2 )}
(d) A.P.

Question. In an AP. the pth term is q and the (p + q)th term is 0. Then the qth term is
(a) – p
(b) p
(c) p + q
(d) p – q

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

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

Question. If a, b, c are in A.P. and a, b, d in G.P., then a, a – b, d – c will be in
(a) A.P.
(b) G.P.
(c) H.P.
(d) None of these

Question. Let a₁, a₂ , a_3.……….. be terms of an A.P. If a₁ + a₂ +……..a_p / a₁ + a₂ +……..a_q = p²/q², p ≠ q , then equals a₆/a₂₁
(a) 41/11
(b) 7/2
(c) 2/7
(d) 11/41

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.

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

Question. The fifth term of the H.P., 2, 2.1/2 , 3.1/3 , ………………. will be

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

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

Question. If A > 0, B > 0 and A + B = π/6 , then the minimum value of tanA + tanB is :
(a) √3 – √2
(b) 4 – 2√3
(c) 2/√3
(d) 2 – √3

Question. The sum of the 3rd and the 4th terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7th term is :
(a) 7290
(b) 640
(c) 2430
(d) 320

Question. Three positive numbers form an increasing G. P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is:
(a) 2 – √3
(b) 2 + √3
(c) √2 + √3
(d) 3 + √2

Question. Let x, y, be positive real numbers such that x + y + = 12 and x³y⁴z⁵ = (0.1) (600)³. Then x³ + y³ + z³ is equal to :
(a) 342
(b) 216
(c) 258
(d) 270

Question. The sum 3/1² + 5/1²+2² + 7/1²+2²+3² + ….. upto 11-terms is:
(a) 7/2
(b) 11/4
(c) 11/2
(d) 60/11

Question. Let f :R → R be a funct ion which sat isfies ƒ(x + y) = ƒ(x) + ƒ(y), ∀ x, y ∈R. If ƒ(a) = 2 and

 then the value of n, for which g(n) = 20, is :
(a) 5
(b) 20
(c) 4
(d) 9

Question. Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of 1/a and 1/b. If 1/M : G is 4 : 5, then a : b can be:
(a) 1 : 4
(b) 1 : 2
(c) 2 : 3
(d) 3 : 4

Question. For x ∈ R, let [x] denote the greatest integer ≤ x, then the sum of the series sum of the series

(a) –153
(b) –133
(c) –131
(d) –135

Question. The sum

 upto 10th term, is :
(a) 680
(b) 600
(c) 660
(d) 620

Question. If a₁, a₂, ………., an are in H.P., then the expression a₁a₂ + a₂a₃ + ………. + a_n–1a_n is equal to
(a) n(a₁ – a_n )
(b) (n -1)(a₁ – a_n )
(c) na₁a_n
(d) (n -1)a₁a_n

Question. If 1 + (1 – 2² · 1) + (1 – 4² · 3) + (1 – 6² · 5) + . . . . . . . + (1 – 20² × 19) = α – 220β, then an ordered pair (α, β) is equal to :
(a) (10, 97)
(b) (11, 103)
(c) (10, 103)
(d) (11, 97)

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

Question. The sum,

is equal to __________.

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

Question.

Question. If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to ___________.

Question. If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then a + b / a – b is equal to :
(a) √6/2
(b) 3√2/4
(c) 7√3/12
(d) 5√6/12

Question. If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 +19 + … is (102)m, then m is equal to:
(a) 20
(b) 25
(c) 5
(d) 10

Question. The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 + ….. upto 11th term is:
(a) 915
(b) 946
(c) 945
(d) 916

Question. The sum of the following series 

(a) 7520
(b) 7510
(c) 7830
(d) 7820

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

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)!

Question. The sum of the first 20 terms of the series 1 + 3/2 + 7/4 + 15/8 + 31/16 + …. is ?
(a) 38 + 1/2²⁰
(b) 39 + 1/2¹⁹
(c) 39 + 1/2²⁰
(d) 38 + 1/2¹⁹

Question. Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² + …… If B – 2A = 100λ , then λ is equal to :
(a) 248
(b) 464
(c) 496
(d) 232

Question. The sum of the series 1 + 4/3 +10/9 + 28/27 + ….. upto n terms is

Question. The sum of the series 1/(1+√2) + 1/(√2+√3) + 1/(3√+√4) + …… upto 15 terms is
(a) 1
(b) 2
(c) 3
(d) 4

Question. Let a, b, c ∈R. If f(x) = ax² + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀ x, y ∈ R, then

 is equal to :
(a) 255
(b) 330
(c) 165
(d) 190

Question. If the sum of the first ten terms of the series

then m is equal to :
(a) 100
(b) 99
(c) 102
(d) 101

Question. For x ∈R, x ≠ -1, if (1 + x)²⁰¹⁶ + x (1 + x)²⁰¹⁵ + x² (1 + x)²⁰¹⁴ + …. + x²⁰¹⁶ =

then a₁₇ is equal to :
(a) 2017! / 17!2000!
(b) 2016! / 17! 1999!
(c) 2016! / 16!
(d) 2017! / 2000!

Question. The sum of the series 1 + 1/4.2! + 1/16.4! + 1/64.6! + …………… ad inf. is
(a) e-1 / √e
(b) e+1 / √e
(c) e-1 / 2√e
(d) e+1 / 2√e

Question. The sum of series 1/2! + 1/4! + 1/6! + …….. is
(a) (e² – 2)/e
(b) (e – 1)²/2e
(c) (e² – 1)/2e
(d) (e² – 1)/2

Question. The sum of first 9 terms of the series. 1³/1 + (1³+2³)/(1+3) + (1³+2³+3³)/(1+3+5) + ……
(a) 142
(b) 192
(c) 71
(d) 96

Question. If

 then k is equal to
(a) 1/6
(b) 17/105
(c) 55/336
(d) 19/112

Question. The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by 10(1/2), then the number of terms in the A.P. is:
(a) 4
(b) 8
(c) 12
(d) 16

Question. If the sum 3/1² + 5/1²+2² + 7/1²+2²+3² + …… + up to 20 terms is equal to k/21, then k is equal to:
(a) 120
(b) 180
(c) 240
(d) 60

Question. If

 then t_n/S_n is equal to
(a) (2n–1)/2
(b) 1/2(n–1)
(c) n–1
(d) (1/2)n

Question. The sum of the series 1/1.2 − 1/2.3 + 1/3.4 …………… up to ∞ is equal to

Question. The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,….., is
(a) 7/81 (179 − 10⁻²⁰)
(b) 7/99 (99 − 10⁻²⁰)
(c) 7/81 (179 − 10⁻²⁰)
(d) 7/9 (99 − 10⁻²⁰)

Question. The value of l² + 3² + 5² + …………………..+ 25² is :
(a) 2925
(b) 1469
(c) 1728
(d) 1456

Question. The sum of the series : 
1 + 1/1+2 + 1/1+2+3 + ……. upto 10 terms, is :
(a) 18/11
(b) 22/13
(c) 20/11
(d) 16/9

Question. If

 where a, b, c are in A.P and |a | < 1, | b | < 1, | c | < 1 then x, y, z are in
(a) G. P.
(b) A.P.
(c) Arithmetic – Geometric Progression
(d) H.P.

Question. If the sum of the roots of the quadratic equation ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then a/c , b/a and c/b are in
(a) Arithmetic – Geometric Progression
(b) Arithmetic Progression
(c) Geometric Progression
(d) Harmonic Progression.

Question.Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + …. + (361 + 380 + 400) is 8000.
Statement-2: 

 for any natural number n.
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
(c) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
(d) Statement-1 is true, statement-2 is false.

Question. If the sum of the series 1² + 2.2² + 3² + 2.4² + 5² + … 2.6² +… upto n terms, when n is even, is n(n+1)² / 2 , then the sum of the series, when n is odd, is
(a) n²(n + 1)
(b) n²(n − 1)/2
(c) n²(n + 1)/2
(d) n²(n − 1)

Question. The sum of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² + …. + 2(2m)² is
(a) m(2m + 1)²
(b) m2(m + 2)
(c) m2(2m + 1)
(d) m(m + 2)²

Question. 1³ – 2³ + 3³ – 4³ + … + 9₃ =
(a) 425
(b) –425
(c) 475
(d) –475

Question. The value of 2^1/4 . 4^1/8 . 8^1/16 … ∞ is 
(a) 1
(b) 2
(c) 3/2
(d) 4

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

Question. The sum of series 1/2! − 1/3! + 1/4! − ……….. upto infinity is
(a) e^½
(b) e^+½
(c) e^-2
(d) e^-1

Question. The sum of the first n terms of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² + …. is n(n+1)² / 2 when n is even. When n is odd the sum is
(a) [ n(n+1) / 2 ]²
(b) n²(n+1) / 2
(c) n(n+1)² / 4
(d) 3n(n+1) / 2

Question. The value of

 is equal to :
(a) 7770
(b) 7785
(c) 7775
(d) 7780

Question. If (10)⁹ + 2(11)¹(10⁸) + 3(11)²(10)⁷ +….. +10(11)⁹ = K(10)⁹, then k is equal to:
(a) 100
(b) 110
(c) 121/10
(d) 441/100

Question.

(a) 199
(b) 99
(c) 200
(d) 19

Question. If the sum of the first n terms of the series √3 + √75 + √243 + v507 +…… is 435√3 , then n equals :
(a) 18
(b) 15
(c) 13
(d) 29

Question. Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
(a) x² -18x -16 = 0
(b) x² -18x +16 = 0
(c) x² +18x -16 =0
(d) x² +18x +16 = 0

Question. Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
(a) 5
(b) 3/5
(c) 8/5
(d) 1/5

Question. If b², a², c² are in A.P., then 1/a + b, 1/b + c, 1/c + a will be in
(a) A.P.
(b) G.P.
(c) H.P.
(d) None of these

Question. If p, q, r are in A.P., a is G.M. between p & q and b is G.M. between q and r, then a², q², b² are in
(a) G.P.
(b) A.P.
(c) H.P
(d) None of these

Question. Sum of the first n terms of the series 1/2 + 3/4 + 7/8 + 15/16 + ……… is equal to :
(a) 2ⁿ – n – 1
(b) 1 – 2^–n
(c) n + 2^–n – 1
(d) 2ⁿ + 1

Question. If 1/b–a + 1/b–c = 1/a + 1/c then a, b, c are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) In G.P. and H.P. both

Question. An infinite G.P has first term x and sum 5, then
(a) x < – 10
(b) – 10 < x < 0
(c) 0 < x < 10
(d) x > 10 C
(d) 4 : 1

Question. If S_n denotes the sum of n terms of a G.P. whose first term is a and the common ratio r, then value of S₁ + S₃ + S₅+ …..+S_2n–1 is

Question. Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is:
(a) 2 – √3
(b) 2 + √3
(c) √3 – 2
(d) 3+ √2

Question. The harmonic mean of a/1–ab and a/1+ab is :
(a) a
(b) a/1–a²b²
(c) 1/1–a²b²
(d) a/1+a²b²

Question. If a, b, c, d, e, f are in A.P., then e – c is equal to:
(a) 2(c – a)
(b) 2(d – c)
(c) 2(f – d)
(d) (d – c)

Question. Let Sn denote the sum of first n terms of an A.P. If S_2n = 3 S_n, then the ratio S_3n/ S_n is equal to :
(a) 4
(b) 6
(c) 8
(d) 10

Question. Let Tr be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m,n, m ≠ n, T_m = 1/n and T_n = 1/m, then a – d equals
(a) 1/m + 1/n 
(b) 1
(c) 1/mn
(d) 0

Question. If the ratio of H.M. and G.M. between two numbers a and b is 4 : 5, then the ratio of the two numbers will be
(a) 1 : 2
(b) 2 : 1
(c) 4 : 1
(d) 1 : 4

Question. Third term of the sequence whose nth term is a_n = 2ⁿ, is
(a) 2
(b) 4
(c) 8
(d) 3