SEQUENCE & PROGRESSION (AP & GP
BOOSTER - A
1) If the difference between two different terms of an AP, with positive common difference is 5 then find the common difference of this AP. 0< d <5
2) If the difference between two terms of an AP is 16 and the common difference is 2, then how many terms are there between two terms ? 7
3) If 4848484848= P(10¹⁰ -1) then determine the value of P. 16/33
4) Two geometric series are t₁, t₂, t₃,.....tₙ and t'₁, t'₂, t'₃,.....t'ₙ. then justify whether the series t₁t'₁ᵐ . t₂t'₂ᵐ . t₃t'₃ᵐ.......tₙt'ₙᵐ are in GP or not. GP
5) Prove that the sum of the first n terms of an arithmetic progression is Sₙ = (n/2) {(Sₙ - Sₙ₋₁)+ S₁}.
6) If the 9th term of GP of a is 2, then determine the product of the first 17 terms of this series. 2¹⁷
7) Which term of the series 2, 4, 8, 16,.... is 384? No terms
8) The term of an AP are t₁, t₂, t₃,.....tₙ, If A₁ = AM(t₁, t₂) , A₂= AM (t₂, t₃),.....etc. then show that A₁, A₂, ......Aₙ₋₁ form an AP. Whose common difference is that of the first series .
9) if the sum of the first four terms of an AP is 4 and the second term is -5 then find the common difference. 12
10) The series a, a², a³,.....is in GP. Then justify whether the series a, √a, ³√a,....is in GP.
11) If 100+ 109+ 118+.....109= p and 599+ 592+ 595+ .....+ 522= q. Find the sum of 622+ 638+.....+13 on terms of p and q. p+ q+ 814
12) If the roots of the equation xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² + ...aₙ=0 are in GP., then show that the product of the first and last terms of this GP is (a₁)²⁾ⁿ (given n is even).
13) If 1+ p+ p²+ p³+......pˣ= (1+ p)(1+ p²)(1+ p⁴) (p> 1) then find the value of x. 7
14) If x, y, z are positive prime integers, show that √x, √y, √z are not any 3 terms, not necessarily consecutive, of an AP.
15) A₁ and A₂ are two arithmetic progression. If the ratio of the sum of first n terms of A₁ and A₂ is (2n +3)/(3n +4), then what is the ratio of their 11th term? 45:67
BOOSTER - B
REASONING TYPE
Each question contains STATEMENT 1 and STATEMENT 2, mark your response according to the following condition
a. if both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1.
b. if both the statements that TRUE but STATEMENT 2 is NOT the correct explanation of STATEMENT 21.
c. if STATEMENT 1 is TRUE and STATEMENT 2 is FALSE .
d. If STATEMENT 1 is FALSE and STATEMENT 2 is TRUE.
1) Statement 1: There are infinite geometric progression for which 27, 8 and 12 are three of its terms(not necessarily consecutive).
STATEMENT 2: Given terms are integers. b
2) Statement 1: x = 1111.......91 times is composite number.
Stetment 2: 91 is composite number. b
3) Statement 1: if an infinite GP has 2nd term x and its sum is 4, then x belongs to (-8,1).
Statement 2: Sum of an infinite GP is finite if for its common ratio r, 0< |r|<1. d
4) Statement 1: 1⁹⁹+ 2⁹⁹ + ........+ 100⁹⁹ is divisible by 10100.
Statement 2: aⁿ + bⁿ is divisible by a+ b if n is odd. a
BOOSTER - C
COMPREHENSIVE TYPE
Comprehensive:
Consider the sequence in the form of groups
(1), (2,2), (3,3,3), (4, 4,4,4),(5, 5,5,5,5),.....
1) The 200th term of sequence is not divisible by
a) 3 b) 9 c) 7 d) none
2) The sum of first 200 terms is
a) 84336 b) 96324 c) 78466 d) none
3) The sum of the remaining terms in the group after 2000th term in which 2000th term lies in
a) 1088 b) 1008 c) 1040 d) none
1d 2a 3b
Let A₁, A₂, A₃,......., Aₘ be the arithmetic means between -2 and 1027 and G₁, G₂, G₃,......Gₙ be the geometric means between 1 and 1024. The product of geometric means is 2⁴⁵ and sum of arithmetic means is 1025 X 171.
4) The value of ⁿᵣ₌₁∑ Gᵣ is
a) 512 b) 2046 c) 1022 d) none
5) The number of arithmetic means is
a) 1243 b) 1542 c) 1330 d) none
6) The number 2A₁₇₁ , G²₅ +1, 2A₁₇₂ are in
a) AP b) GP c) HP d) none
4c 5b 6a
Two arithmetic progression have the same number. The ratio of the last term of the first progression to first term of the second progression is equal to the ratio of the last term of the second progression to the first term of the first progression and is equal to 4, the ratio of the sum of the n terms of the first progression to the sum of the n terms of the second progression is equal to 2.
7) The ratio of their common difference is
a) 12 b) 24 c) 26 d) 9
b) The ratio of their n-th term is
a) 6/5 b) 7/2 c) 9/5 d) none
c) Ratio of their first term is
a) 2/7 b) 3/5 c) 4/7 d) 2/5
7c 8b 9a
BOOSTER - D
1) The sum of integers from 1 to 100 that are divisible by 2 or 5 is
a) 3000 b) 3050 c) 4050 d) none
2) If the 9th term of an AP is 35 and 19th is 75, then its 20th terms will be
a) 78 b) 79 c) 80 d) 81
3) The 9th term of the series 27+9+ 27/5+ 27/7+......will be
a) 27/17 b) 10/17 c) 16/27 d) 17/27
4) If the p-th, q-th and r-th term of an AP are a, b and c respectively, then the value of a(q - r)+ b(r - p) + c(p - q)=
a) -1 b) 1 c) 0 d) 1/2
5) if n-th terms of two AP 's are in 3n+8 au7n+ 15, then the ratio of their 12th terms will be
a) 4/9 b) 7/16 c) 3/7 d) 8/15
6) Let tᵣ be the rᵗʰ term of an AP for= 1,2,3....if for some positive integers we have tₘ = 1/n and tₙ = 1/m, then tₘₙ equals
a) 1/mn b) 1/m + 1/n c) 1 d) 0
7) If the ratio of the sum of terms of two AP's be (7n +1): (4n + 27), then the ratio of their 11th terms will be
a) 2:3 b) 3:4 c) 4:3 d) 5:6
8) The sum of the series 1/2+ 1/3+1/6+.....to 9 terms is
a) -5/6 b) -1/2 c) 1 d) -3/2
1b 2b 3a 4c 5a 6c 7c 8d
BOOSTER - E
1) The interior angles of a polygon are in AP. If the smallest angle between 120° and the common difference between 5, then the number of sides is
a) 8 b) 10 c) 9 d) 6
2) If the sum of the series 2+5+8+11......is 60100, then the number of terms are
a) 100 b) 200 c) 150 d) 250
3) The sum of all natural numbers between 1 and 100 which are multiples of 3 is
a) 1680 b) 1683 c) 1681 d) 1682
4) If the sum of the series 54+51+48+......is 513, then the number of terms are
a) 18 b) 20 c) 17 d) none
5) If the sum of n terms of AP is 2²+ 5, then the n-th term will be
a) 4+3 b) 4+5 c) 4+6 d) 4+7
6) The maximum sum of the series 20+19+18+......is
a) 310 b) 300 c) 320 d) none
7) The sum of the numbers between 100 and 1000 which is divisible by 9 will be
a) 55350 b) 57228 c) 97015 d) 62140
8) if denotes the sum of terms of an arithmetic progression, then the value of ( ) is equals to (S₂ₙ - Sₙ) is equal to
a) 2Sₙ b) S₃ₙ c) 1/3 d) (1/2)Sₙ
9) The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is
a) 2489 b) 4735 c) 2317 d) 2632
1c 2b 3b 4a 5a 6a 7a 8c 9d
BOOSTER - F
1) Let Sₙ denotes the sum of n terms of an AP S₂ₙ = 3 Sₙ, then ratio S₂ₙ/Sₙ =
a) 4 b) 6 c) 8 d) 10
2) the sum of the first four terms of an AP is 56. The sum of the last four times is 112. If the first term is 11, the number of terms is
a) 10 b) 11 c) 12 d) none
3) The sum of the series x/(1- x²) + x²/(1- x⁴)+ x⁴/(1- x⁸)+......to infinite terms, if |x|< 1 is
a) x/(1- x) b) 1/(1- x) c) (1+ x)/(1- x) d) 1
4) if a, b, c , d are positive real number such that a+ b+ c+ d=2, then M= (a+ b)(c + d) satisfies the relations:
a) 0< M ≤ 1 b) 1<M ≤2 c) 2≤M ≤3 d) 3≤M ≤ 4
5) If a₁, a₂, ....., aₙ are positive real numbers whose products a fixed number c, then the minimum value of a₁ + a₂ +......+ aₙ₋ₐ + 2aₙ is
a) n(2c)¹⁾ⁿ b) (n +1)c¹⁾ⁿ c) 2nc¹⁾ⁿ d) (n +1)(2c)¹⁾ⁿ
6) Suppose a,b,c are in AP and a², b², c² are in GP. If a< b < c and a+ b + c = 3/2, then the value of a is
a) 1/2√2 b) 1/2√3 c) 1/2 - 1/√3 d) 1/2 - 1/√2
7) If a₁, a₂, a₃,.......a₂ₙ₊₁ are in AP then
(a₂ₙ₊₁ - a₁)/(a₂ₙ₊₁ + a₁) + (a₂ₙ - a₂)/(a₂ₙ + a₂) +.......+ (aₙ₊₂ - aₙ)/(aₙ₊₂ + aₙ) is equal to
a) n(n+1)/2. (a₂ -a₁)/aₙ₊₁
b) n(n+1)/2
c) (n +1)(a₂ - a₁) d) none
8) If a₂a₃/a₁a₄ = (a₂ + a₃)/(a₁+ a₄) = 3(a₂ - a₃)/(a₁ - a₄) then a₁, a₂, a₃, a₄ are in
a) AP b) GP c) HP d) none
9) If (a + beʸ)/(a - beʸ) = (b - ceʸ)/(b - ceʸ) = (c + deʸ)/(c - deʸ), then a b,c, d are in
a) in AP b) in GP c) in HP d) equal
10) If a,b,c are in GP, x and y be the AM's between a,b, and b, c respectively, then (a/x + c/y)(b/x + b/y) is equal to
a) 2 b) -4 c) 3 d) 4
11) Let Sₖ = ₙl→∞ ⁿ ᵢ₌₀∑ 1/(k+1)ⁱ. Then ⁿₖ₌₁∑ kSₖ equals
a) n(n+1)/2 b) n(n -1)/2 c) n(n +2)/2 d) n(n +3)/2
12) In the given square, a diagonal is drawn, and equally spaced parallel line segments joining points on the adjacent sides are drawn on both sides of the diagonal. The length of the diagonal is n √2 cm. If distance between consecutive line segments be 1/√2 cm, then the sum of the lengths of all possible line segments and the diagonals is
a) n(n +1)√2 cm b) n² cm c) n(n +2) cm d) n²√2 cm
13) Sum to infinite terms of the series 1/(1.3) + 2/(1.3.5) + 3/(1.3.5.7) + 4/(1.3.5.7.9) +.....is
a) 1 b) 1/2 c) 3/2 d) none
14) If 1⁴/(1.3) + 2⁴/(3.5) + 3⁴/(5.7) +.....+ n⁴/{(2n -1)(2n +1)} = (1/48) f(n)+ n/{16(2n +1)}. Then f(n) is equal to
a) n(n+1)(2n +1)
b) n(4n²+1)
c) n(4n²+6n +5) d) none
1b 2b 3a 4a 5a 6d 7a 8c 9b 10d 11d 12d 13b 14c
BOOSTER - G
1) a₁, a₂, a₃, a₄ are in HP then 1/(a₁a₄) ³ᵣ₌₁∑aᵣ aᵣ₊₁ is a root of
a) x²+ 2x +15=0
b) x²+ 2x -15=0
c) x²- 6x -8=0
d) x²- 9x + 20=0
2) If x¹⁵ - x¹³ + x¹¹ - x⁹+ x⁷ - x⁵ + x³ - x =7, where x > 0, then
a) x¹⁶ is equal to 15
b) x¹⁶ is less than 15
c) x¹⁶ is greater than 15
d) Nothing can be said regarding the value of x¹⁶
3) The sum to n terms of the series 2/(2.3) + (2².2)/(3.4) + (2².3)/(4.5)+....is
a) 2ⁿ⁻¹/(n +2)
b) 2ⁿ⁻¹ - (n +1)/(n +2)
c) 2ⁿ⁺¹/(n +2) - 1
d) none
4) n arithmetic means are inserted between the numbers 7 and 49. If sum of these be 364, then the sum their squares is
a) 10380 b) 11380 c) 11830 d) 18130
5) If a,b,c are positive numbers such that a+ b+ c=1, then the minimum value of 1/ab + 1/bc + 1/ca is
a) 3 b) 9 c) 27 d) none
6) The sum of the series 9/(5².2.1) + 13/(5².3.2) + 17/(5⁴.4.3)+.....up to infinite terms is equal to
a) 1 b) 9/5 c) 1/5 d) 2/5
7) The series 8/5 + 16/65 + 24/325+......up to infinity has the sum equal to
a) 16/5 b) 2 c) 4 d) 5
8) The sum of the series ᵣʳ₌₀∑ (-1)ʳ (n + 2r)², (where is Even) is equal to
a) - n²+ 2n
b) - n²+ 4n
c) - n²+ 3n
d) - 4n²+ 2n
1b 2c 3c 4c 5c 6c 7b 8d
BOOSTER- H
1) The positive integral values of n such that
1.2¹+ 2.2² + 3.2³+4.2⁴+5.2⁵+....+n.2ⁿ= 2ⁿ⁺¹⁰ +2 is
a) 313 b) 513 c) 413 d) 613
2) The value of ∞ᵣ₌₁∑ {r³+ (r² +1)³}/{r⁴+ r² +1)(r²+ r)} is
a) 3/2 b) 1/2 c) 0 d) infinite
3) If log(a+ c), log(a+ b), log(b + c) are AP and a,c,b are in HP, then the value of a+ b is (given a,b,c > 0)
a) 2c b) 3c c) 4c d) 6c
4) Sum Sₙ = 1+5+19+65+.....
a) (3ⁿ⁺¹- 2ⁿ⁺¹ -1)/2
b) (3ⁿ⁺¹ - 2ⁿ⁺² +1)/2
c) (3ⁿ⁺¹ + 2ⁿ⁺¹ -1)/2
d) (3ⁿ⁺² - 2ⁿ⁺³ +1)/2
5) If a₁, a₂, a₃ , .....aₙ are in AP with Sₙ as the sum of first n terms, then ⁿₖ₌₀∑ⁿCₖ Sₖ is equal to
a) 2ⁿ⁻²(na₁ + Sₙ)
b) 2ⁿ(a₁ + Sₙ)
c) 2(na₁ + Sₙ)
d) 2ⁿ⁻¹(a₁ + Sₙ)
1b 2a 3a 4b 5a
BOOSTER - I
Comprehensive -1
Suppose a series of n terms is given by Sₙ = t₁ + t₂ + t₃ + .....tₙ
Then Sₙ₋₁ = t₁ + t₂ + t₃+......+tₙ₋₁ , n > 1. Subtracting we get, Sₙ - Sₙ₋₁ = tₙ, nm≥ 2
Further if we put n=1 in the first sum then S₁ = t₁, Thus we can write tₙ = Sₙ - Sₙ₋₁ , n≥ 2 and t₁ = S₁
1) The sum of n terms of a series is a.2ⁿ - b, where a and b are constants then the series is
a) AP b) GP c) AGP d) GP from second term onwards
2) The sum of n terms of a series is a.2ⁿ - b, then the sum ∞ᵣ₌₂∑ 1/tᵣ is
a) a b) a/2 c) 1/a d) 2/a
3) The sum of n terms of a series is given by (1/4) n(n+1)(n +2)(n +3) then the n-th term of the series is
a) n(n+1)(n+2)
b) (n+1)(n+2)(n +3)
c) n(n+1)(2n+1)/6
d) n(4n²-1)
Comprehension - 2
We know that, if a₁, a₂,......aₙ are in HP, then 1/a₁ , 1/a₂ , ......, 1/aₙ, are in AP and vice-versa. If a₁, a₂,.....aₙ are in AP with common difference d, then for any b(> 0), the number bᵃₐ , bᵃ₂, bᵃ₃,.....bᵃₙ are in GP with common ratio bᵈ. If a₁, a₂,......aₙ are positive and in GP with common ratio r, then for any base b(> 0), logᵥa₁, log ᵥa₂,.....logᵥaₙ are in AP with common difference logᵥr.
(Note v as r and aa,a2,a3 as aₐ, a₂, a₃)
4) If x, y, z are respectively the pᵗʰ, qᵗʰ and the rᵗʰ terms of an AP as well as of a GP, then the value of xʸ⁻ᶻ, yᶻ⁻ˣ, zˣ⁻ ʸ is
a) 1 b) -1 c) 0 d) 2
5) If a,b,c,d are in GP and aˣ = bʸ = cᶻ = dᵛ, then x, y, z, v are in
a) AP b) GP c) HP d) none
6) If a, b, c are in HP, then ₄-a⁻¹, ₄-b⁻¹, ₄-c⁻¹ are in
a) AP b) GP c) HP d) none
1d 2c 3a 4a 5c 6b
BOOSTER - J
Each of these questions has 4 choices (a), (b), (c) and (d) for its answer, out of which ONE Or MORE is/are correct.
1) If the first and (2n -1)th terms of an AP, a GP and HP of positive terms are equal and their n-th terms are a, b, c respectively, then
a) a= b = c b) a≥ b ≥ c c) b²= ac d) a+ c= 2b
2) For a positive integer n, let a(n)= 1+ 1/2+ 1/3 + 1/4+.....+ 1/{(2ⁿ) -1}. Then
a) a(100)< 100
b) a(200)< 200
c) a(200)> 100
d) a(2010)< 1005
3) If x, y, z are positive numbers in AP, then
a) y²≥ xz b) xy+ yz ≥ 2xz c) (x + y)/(2y - x) + (y + z)/(2y - z) ≥ 4 d) none
4) All the terms of an AP are natural numbers and the sum of the first 20 terms is greater than 1072 and less than 1162. If the sixth term is 32 then
a) first term is 12
b) first term is 7
c) common difference is 4
d) common difference is 5
5) If (m+ 1)ᵗʰ, (n +1)ᵗʰ and (r +1)th terms of an AP are in GP and m,n,r are in HP, then the ratio of the first term of the AP its common difference is
a) -n/2 b) -m/2 c) r d) -mr/(m + r)
6) If a₁, a₂, .....aₙ are in AP with common difference d, then
cot ⁻¹{(1- a₁a₂)/d} + cot⁻¹{(1+ a₂a₃)/d} + cot⁻¹{(1+ a₃a₄)/d} + .....+ cot⁻¹{(1+ aₙ aₙ₋₁)/d} is
a) tan⁻¹aₙ - tan⁻¹a₁
b) cot⁻¹a₁ + cot⁻¹aₙ
c) cot⁻¹a₁ - cot⁻¹aₙ d) none
7) ( 666....6 ) ( 888....8 ) is equal to
ⁿ ᵈᶦᵍᶦᵗˢ ⁿ ᵈᶦᵍᶦᵗˢ
a) (9/4) (10ⁿ -1)
b) (4/9) (10²ⁿ-1)
c) (4/9) (10ⁿ -1) d) none
8) If x= ∞ ₙ₌₀ ∑ aⁿ , y= ∞ ₙ₌₀ ∑ bⁿ , z= ∞ ₙ₌₀ ∑ (ab)ⁿ , where |a|, |b|< 1, then
a) xyz= x + y + z
b) xy + yz= x y + z
c) xy + yz= xz + y d) none
9) If Sᵣ denotes the sum of the first r terms of an AP then (S₃ᵣ - Sᵣ₋₁)/(S₂ᵣ - S₂ᵣ₋₁)=
a) 2r + 1
b) 2r - 1
c) 2r + 3 d) none
10) Let α, β, γ be the roots of the equation 3x³- x²- 3x +1=0. If α, β are in HP, then |α - γ|=
a) 1/3 b) 2/3 c) 4/3 d) none
11) Suppose a, b > 0 and x₁, x₂, x₃ (x₁ > x₂ > x₃) are roots of (x - a)/b + (x - b)/a = b/(x - a) + a/(x - b) and x₁ - x₂ - x₃ = c, then a,b,c are in
a) AP b) GP c) HP d) none
12) If 0.272727...... , x and 0.727272......are in HP, then x must be
a) rational b) integer c) irrational d) none
13) In a geometric series, the first term is a and common ratios is r. If Sₙ denotes the sum of n terms and Uₙ = ⁿₙ₌₁∑ Sₙ then rSₙ + (1- r)uₙ =
a) na b) (n -1)a c) (n +1)a d) none
1bc 2abcd 3abc 4bd 5ad 6ac 7b 8b 9a 10c 11c 12a 13a
BOOSTER- K
²²²²²₁₂₃₁₂ₚ₁₂q₆₂₁ₙₙₓₓₚ³²²³²²ˣʸᶻᵥₐ꜀ₐᵥ꜀²²²²²⁴⁵¹⁰¹⁵⁵⁰ᵗʰᵗʰᵗʰ²₁₂₂₀₁₂₂₀¹⁹ᵣ₌₁ᵣᵣ₊₁²³ₙ₁₂₃₂₀₀₃∞²²²²ₙ₊₁ₙ₁₃²⁰⁰ᵣ₌₁∑ᵣ²₁²₂²ₙₙ²²ₙₙⁿₙ₌₁∑ₙₙₙ₁₂ₙ₁₁₂₂ₙₙ₁₂ₙ₁₁₂₂ₙₙᵣᵣₜᵗʰᵗʰₙₙₙ₋₁²ₘ²²²²²²²
ⁿ₌₁²⁰⁰¹ⁿᵣ₌₁ₙ⁵³⁴²ₙ²ₙ²ₙₙₙₙₙₘ₌₁ₘ₌₁ ²ᵐⁿᵢ₌₀ⱼ₌₀ₖ₌₀ ᶦʲᵏₙₙₙₙ₊₁ₙₙ₊₁ₐ⁻¹ᵥ⁻¹₁₂₃₂ₙ₊₁₂ₙ₊₁₁₂ₙ₊₁₁₂ₙ₁₂ₙ₁ₙ₊₂ₙ₊₂ₙ₂₁ₙ₊₁₂₁ₙ ⁿ²ⁿⁿₙ₌₀ⁿₙ₌₀ₙ₌₀ⁿᵣ₃ᵣ₋₁₂ᵣ₂ᵣ₋₁αβγ³²αβαγ₁₂₃₁₂₃₁₂₃ⁿⁿ⁴⁴⁴∞⁴⁴⁴⁴∞⁴⁴⁴ₙₙₙₙₙ₁₂₃₃₁₁₂₂₃₁₂₃₃₁₁₂₂₃₁₂₃₃₁₁₂₂₃₁₂₃ᵣᵗʰₘₙₘₙ₁₂₁₀ ₁₂₁₀₁₁₁₀₁₀₄₇²ₙₙ₊₁ₙ²²²αβ²²³³²²²²²²²²²²
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