Maths MCQ Class 11 Ch-8 | Sequence & Series

 MCQ | Class 11 | Ch-8 SEQUENCE & SERIES

Topics to be covered here

MCQ Based on the ARITHMATIC PROGRESSION (AP).

MCQ Based on the GEOMETRIC PROGRESSION (GP)

MCQ Based on the ARITHMATIC MEAN (AM) AND GEOMETRIC MEAN (GM).

MCQ | CHAPTER 8 | CLASS 11
SEQUENCE AND SERIES.

Q 1) If a, b, c are in AP then
(a) b = a + c
(b) 2b = a + c
(c) b²  = a + c
(d) 2b²  = a + c
Ans: b

Q 2) What is  the n^th term  of an A.P.?
a) a_n  = a + (n - 1) d
b) a_n  = a + (n) d
c) a_n = ar^n-1   
d) a_n = arⁿ  

Ans: a 

Q 3) If an A.P. is 3, 5, 7, 9 ……. Find the 12^th term of the A.P.
a) 12        b) 21        c) 22        d) 25 

Ans: d

Q 4) If general term of an A.P. is 3n then find common difference.
a) 2        b) 3        c) 5        d) 6 

Ans: b

Q 5) In A.P. 171, 162, 153, ………. Find first negative term.
a) 0        b) - 2        c) - 6        d) – 9 

Ans: d

Q 6) Insert 4 numbers between 2 and 22 such that the resulting sequence is an A.P.
a) 4, 8, 12, 16
b) 5, 9, 13, 17
c) 4, 10, 15, 19
d) 6, 10, 14, 18 

Ans: d

Q 7) If two numbers are 2 and 6 then find their arithmetic mean.
a) 3        b) 4        c) 5        d) 8 

Ans: b

Q 8) The sum of n terms of two arithmetic progressions are in the ratio (2n - 7) : (7n + 5). Find the ratio of their 9^th terms.
a) 4 : 5        b) 5 : 4        c) 9 : 31        d) 31 : 9 

Ans: c

Q 9) If in an A.P., first term is 20, common difference is 2 and nth term is 42, then find n.
a) 10        b) 11        c) 12        d) 14 

Ans: c

Q 10) If 1/(b + c), 1/(c + a), 1/(a + b) are in AP then
(a) a, b, c are in AP
(b) a², b², c² are in AP
(c) 1/1, 1/b, 1/c are in AP
(d) None of these
Ans: b

Solution
Given, 1/(b + c), 1/(c + a), 1/(a + b)
⇒ 2/(c + a) = 1/(b + c) + 1/(a + b)

⇒ 2b² = a² + c²
⇒ a², b², c² are in AP

Q 11) The sum of AP 2, 5, 8, ….. up to 50 terms is
(a) 3557        (b) 3775        (c) 3757        (d) 3575
Ans: b
Solution
Given, AP is 2, 5, 8, …..up to 50
Now, first term a = 2
common difference d = 5 – 2 = 3
Number of terms = 50
Now, Sum = (n/2) × {2a + (n – 1)d}
= (50/2) × {2 × 2 + (50 – 1)3}
= 25 × {4 + 49 × 3}
= 25×(4 + 147)
= 25 × 151 = 3775

Q 12) If 2/3, k, 5/8 are in AP then the value of k is
(a) 31/24        (b) 31/48        (c) 24/31        (d) 48/31
Ans: b 

Solution

Given, 2/3, k, 5/8 are in AP
⇒ 2k = 2/3 + 5/8
⇒ 2k = 31/24
⇒ k = 31/48
So, the value of k is 31/48

Q 13)  If the third term of an A.P. is 7 and its 7^th  term is 2 more than three times of its third term, then the sum of its first 20 terms is
(a) 228        (b) 74        (c) 740        (d) 1090
Ans: c
Solution

Let a is the first term and d is the common difference of AP
Given the third term of an A.P. is 7 and its 7th term is 2 more than three times of its third term
⇒ a + 2d = 7 ………….. (1)
and
3(a + 2d) + 2 = a + 6d
⇒ 3×7 + 2 = a + 6d
⇒ 21 + 2 = a + 6d
⇒ a + 6d = 23 ………….. (2)
From equation 1 – 2, we get
4d = 16 ⇒ d = 16/4 ⇒ d = 4
From equation 1, we get
a + 2×4 = 7 ⇒ a + 8 = 7 ⇒ a = -1
Now, the sum of its first 20 terms
= (20/2) × {2 × (- 1) + (20 - 1) × 4} = 10 × {- 2 + 19 × 4)}
= 10 × {- 2 + 76)} = 10 × 74 = 740

Q 14) If the sum of the first 2n terms of the A.P. 2, 5, 8, ….., is equal to the sum of the first n terms of the A.P. 57, 59, 61, ….., then n equals
(a) 10        (b) 12        (c) 11        (d) 13
Ans: c 

Explanation

Given, the sum of the first 2n terms of the A.P. 2, 5, 8, …..= the sum of the first n terms of the A.P. 57, 59, 61, ….
⇒ (2n/2) × {2 × 2 + (2n - 1) 3} = (n/2) × {2 × 57 + (n - 1)2}
⇒ n × {4 + 6n – 3} = (n/2) × {114 + 2n – 2}
⇒ 6n + 1 = {2n + 112}/2
⇒ 6n + 1 = n + 56
⇒ 6n – n = 56 – 1
⇒ 5n = 55
⇒ n = 55/5
⇒ n = 11

Q 15)  If 7^th and 13^th terms of an AP  be 34 and 64 respectively, then its 18^th term is
a) 87        b) 88        c) 89        d) 90 

Ans: c

Q 16) If sum of p terms of an AP is q and sum of q terms is p then what is the sum of p + q terms
a) 0        b) p - q        c) p + q        d –(p + q) 

Ans: d

Solution Hint: Find S_p = q and S_q = p and then subtract these two

Q 17) If the sum of n terms of an AP be 3n² - n and its common difference is 6, then its first term is
a) 2        b) 3        c) 1        d) 4 

Ans: a

Q 18) The first and last term of an AP is 1 and 11. If the sum of its term is 36, then the number of terms will be
a) 5        b) 6        c) 7        d) 8 

Ans: b

Q 19) If sum of its first n terms of an AP is 3n² + 5n then which of its term is 164 ?
a) 26^th        b) 27^th        c) 28^th        d) None of these 

Ans: b
Q 20)   If 3^rd term of an A.P. is 6 and 5^th term of that A.P. is 12. Then find the 21^st term of that A.P.

a) 40        b) 42        c) 60        d) 63 

Ans: c

Q 21) If in an A.P., first term is 20 and 12^th term is 120. Find the sum up to 12^th term.
a) 420        b) 840        c) 140        d) 1680 

Ans: b

Q 22) If an A.P. is 1,7,13, 19 , ……… Find the sum of 22 terms.
a) 127        b) 1204        c) 1408        d) 1604 

Ans: c

Q 23) If sum of n terms of an A.P. is n² + 5n then find general term.
a) n + 1
b) 2n + 4
c) 3n
d) n² + 3n
Ans:  b

Q 24) If n AM’s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
a) 6        b) 8        c) 4            d) None of these 

Ans: a

Q 25) If a constant is added or subtracted from each term of an A.P. then resulting sequence is also an A.P.
a) True
b) False 

Ans: a

Q 26) If a constant is multiplied to A.P. then resulting sequence is also an A.P.
a) True
b) False 

Ans: a

MCQ Based on Geometric Progression

Q 27)    If first term of a G.P. is 20 and common ratio is 4. Find the 5^th term.
a) 10240        b) 40960        c) 5120            d) 2560 

Ans: c

Q 28) The third term of a geometric progression is 4. The product of the first five terms is

(a) 4³          (b) 4⁵             (c) 4⁴         (d) none of these 

Ans: b

Q 29) The first term of a GP is 1. The sum of the third term and fifth term is 90. The common ratio of GP is
a) 1        b) 2        c) 3        d) 4 

Ans: c

Q 30) If a sequence is in the form 2x5ⁿ then which of the following may be the sequence?
a) Arithmetic progression
b) Geometric Progression
c) Harmonic Progression
d) Special Progression 

Ans: b

Q 31) What is n^th term of a G.P.?

a) a_n = a + (n-1) d                                         

b) a_n = a + (n) d
c) a_n = ar^n-1                                                    

d) a_n = arⁿ

Ans:  c

Q 32) If r=1 in a G.P. then what is the sum to n terms?
a) na
b) a/ n
c) (n - 1) a
d) (n + 1) a
Ans: a
Explanation

If a is the first term of G.P., then G.P. look like a, a, a, a,  …………
Then sum to n terms becomes na.

Q 33)   If a = 3 and r = 2 then find the sum up to 5^th term.
a) 95        b) 82            c) 93        d) 97 

Ans: c

Q 34)   In G.P. 4, 8, 16, 32, ………… find the sum up to 5^th term.
a) 16        b) 64        c) 128        d) 124 

Ans: d

Q 35) Find the sum of series 1+1/2 + 1/4 + ………. up to 6 terms.
a) 63/32        b) 32/63        c) 26/53        d) 53/26 

Ans: a

Q 36) In a G.P., 5^th term is 27 and 8^th term is 729. Find its 11^th term.
a) 729        b) 2187        c) 6561        d) 19683 

Ans: d

Q 37) How many terms of G.P. 2, 4, 8, 16, …………… are required to give sum 254?
a) 4        b) 5            c) 6        d) 7 

Ans: d

Q 38) The sum of first three terms of a G.P. is 21/2 and their product is 27. Find the common ratio.
a) 2
b) 1/ 2
c) 2 or 1/ 2
d) neither 2 nor 1/2

Ans: c

Q 39) The sum of first three terms of a G.P. is 21/2 and their product is 27. Which of the following is not a term of the G.P. if the numbers are positive?
a) 3        b) 2/3        c) 3/2        d) 6
Ans: b
Explanation: 

Let three terms be: a/r, a, ar.
Product = 27
⇒ (a/r) (a) (ar) = 27

⇒ a³ = 27 ⇒ a = 3
Sum = 21/ 2
⇒ (a / r + a + ar) = 21/2
⇒ a (1 / r + 1 + 1r) = 21/2
⇒ 3 (1 / r + 1 + 1r) = 21/2
⇒ (1 / r + 1 + 1r) = (21/2)/3 = 7/2

⇒ (r² + r + 1) = (7/2) r        

⇒ r² – (5/2) r +1 = 0
⇒ r = 2 and 1/2.
Terms are 3/2, 3, 3 x 2 i.e. 3/2, 3, 6.

Q 40) Which of the following is the geometric mean of 3 and 12 ?
a) 4        b) 6        c) 9        d) 10 

Ans: b

Q 41) If three positive numbers are inserted between 4 and 512 such that the resulting sequence is a G.P., which of the following is not among the numbers inserted?
a) 256        b) 16        c) 64        d) 128
Ans: a

Explanation:

Let G.P. be 4, G₁, G₂, G₃, 512.
⇒ a = 4 and t₅ = ar⁴ = 512 (Given)
⇒ 4 x r⁴ = 512
⇒ r⁴ = 512/4 = 128 ⇒ r = 4.
G₁ = a₂ = a r = 4 x 4 = 16.
G₂ = G₁ x r = 16 x 4 = 64.
G₃ = G₂ x r = 64 x 4 = 256.

Q 42) If A.M. of two numbers is 15/2 and their G.M. is 6, then find the two numbers.
a) 6 and 8        b) 12 and 3            c) 24 and 6        d) 27 and 3
Ans: b
Explanation: 

We know, A.M. of two numbers a and b is (a + b)/2
⇒ (a + b)/2 = 15/2 ⇒ a + b = 15.
Also, G.M. of two numbers a and b is
⇒   = 6 ⇒ ab = 36.

⇒ a(15 - a) = 36 ⇒ a = 3 or 12.
For a = 3, b = 12.
For a = 12, b = 3.
So, the two numbers are 3 and 12.

Q 43) Which of the following is true if A means arithmetic mean and b means geometric mean of two numbers?
a) A > G
b) A ≥ G
c) G < A
d) G ≤ A 

Ans: b

Q 44) Find the sum 1² + 2² + 3² + …………… + 10².
a) 325        b) 365        c) 385        d) 435 

Ans: c

Q 45) Find the sum 1³ + 2³ + 3³ + …………… + 8³.
a) 1225        b) 1184        c) 1475        d) 1296 

Ans: d

Q 46) Find the sum of series up to 6^th term whose n^th term is given by n² + 3ⁿ.
a) 91        b) 1284        c) 1183        d) 1092 

Ans: c

Q 47) Find the sum to 6 terms of each of the series 2 x 3 + 4 x 6 + 6 x 11 + 8 x 18 + ………………..
a) 784        b) 882        c) 928        d) 966 

Ans: d

Q 48) Find the sum up to 7th term of series 2 + 3 + 5 + 8 + 12 + ………………….
a) 70        b) 490        c) 340        d) 420 

Ans: a

Q 49) Find the sum of series 6²+7²+…………………..+15².
a) 55            b) 1185        c) 1240           d) 1385 

Ans: b

Q 50) Find the sum of series 6³+7³+ ………………..…..+20³.
a) 43875        b) 83775        c) 43775        d) 43975 

Ans: a

Q 51) Find the sum of series 1² + 3² + 5² + ………………………….. + 11².
a) 279        b) 286        c) 309        d) 409 

 Ans: b

Q 52) The ratio of the A.M. and G.M. of two positive numbers a and b is 5: 3. Find the ratio of a to b.
a) 9 : 1        b) 3 : 5        c) 1 : 9        d) 3 : 1
Ans: a 

Explanations

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