Sequences and Series

The positive integer n, for which the solutions of the equation x(x + 2) + (x + 2)(x + 4) + + (x + 2n - 2)(x + 2n) = 8n3 are two consecutive even integers, is :

1. 3
2. 6
3. 9
4. 12

Let a1, a22, a322, , a1029 be a G.P. of common ratio 1 2. If a1 + a2 + + a10 = 62, then a1 is equal to:

1. 2 - 1
2. 2( 2 - 1)
3. 2 - 2
4. 2(2 - 2)

If 114 + 124 + 134 + = 490, 114 + 134 + 154 + = , 124 + 144 + 164 + = , then is equal to :

1. 23
2. 14
3. 18
4. 15

Let x1, x2, x3, x4 be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x1, x2, x3, x4, then the resulting numbers are in an arithmetic progression. Then the value of 124(x1 x2 x3 x4) is:

1. 18
2. 216
3. 36
4. 72

Let an be the nth term of an A.P. If Sn = a1 + a2 + a3 + + an = 700, a6 = 7 and S7 = 7, then an is equal to :

1. 65
2. 56
3. 70
4. 64

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :

1. 757
2. 755
3. 750
4. 760

1 + 3 + 52 + 7 + 92 + up to 40 terms is equal to

1. 40870
2. 41880
3. 43890
4. 33980

If the sum of the first 20 terms of the series 4 14 + 3 12 + 14 + 4 24 + 3 22 + 24 + 4 34 + 3 32 + 34 + 4 44 + 3 42 + 44 + is mn, where m and n are coprime, then m + n is equal to :

1. 423
2. 421
3. 422
4. 420

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of A.P.'s in A and B respectively such that D=d+3, d>0. If p+qp-q= 195, then p-q is equal to

1. 540
2. 450
3. 600
4. 630

Let a1, a2, a3, be a G.P. of increasing positive numbers. If a3 a5 = 729 and a2 + a4 = 1114, then 24(a1 + a2 + a3) is equal to

1. 128
2. 129
3. 131
4. 130

The sum 1 + 3 + 11 + 25 + 45 + 71 + upto 20 terms, is equal to

1. 7240
2. 8124
3. 7130
4. 6982

The sum 1 + 1 + 32! + 1 + 3 + 53! + 1 + 3 + 5 + 74! + up to terms, is equal to

1. 3e
2. 2e
3. 4e
4. 6e

Let a1, a2, a3, be in an A.P. such that k=112 a2k-1 = - 725a1, a1 0. If k=1n ak = 0, then n is :

1. 18
2. 17
3. 11
4. 10

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by 212. Then the number of terms which are integers in the A.P. is :

1. 6
2. 4
3. 8
4. 10

If the sum of the first 10 terms of the series 4 11 + 4 14 + 4 21 + 4 24 + 4 31 + 4 34 + is mn, where (m, n) = 1, then m + n is equal to .

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is :

1. 108
2. 90
3. 122
4. 84

The value of n ( k=1n k3 + 6k2 + 11k + 5(k+3)! ) is :

1. 5/3
2. 2
3. 4/3
4. 7/3

Let a1,a2, ,a2024 be an Arithmetic Progression such that a1+(a5+a10+a15+ +a2020)+a2024=2233. Then a1+a2+a3+ +a2024 is equal to .

Let an be a sequence such that a0 = 0, a1 = 12 and 2an+2 = 5an+1 - 3an, n = 0, 1, 2, 3, . Then k=1100 ak is equal to

1. 3a100 + 100
2. 3a100 - 100
3. 3a99 - 100
4. 3a99 + 100

Let Tr be the rth term of an A.P. If for some m, Tm = 125, T25 = 120, and 20 r=125Tr = 13, then 5m r=m2mTr is equal to

1. 98
2. 126
3. 112
4. 142

For positive integers n, if 4an = (n2 + 5n + 6 ) and Sn = k=1n ( 1ak ), then the value of 507S2025 is :

1. 540
2. 675
3. 1350
4. 135

The interior angles of a polygon with n sides are in an A.P. with common difference 6 . If the largest interior angle of the polygon is 219 , then n is equal to .

Let Sn= 12+ 16+ 112+ 120+ upto n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is 2026 S2025, then the absolute difference between 20th and 15th terms of the A.P. is

1. 20
2. 45
3. 90
4. 25

In an arithmetic progression, if S40=1030 and S12=57, then S30-S10 is equal to :

1. 525
2. 505
3. 510
4. 515

If 7=5+ 17(5+ )+ 172(5+2 )+ 173(5+3 )+ , then the value of is :

1. 17
2. 1
3. 67
4. 6

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

1. -120
2. -1200
3. -1080
4. -1020

The roots of the quadratic equation 3x2-px+q=0 are 10th and 11th terms of an arithmetic progression with common difference 32. If the sum of the first 11 terms of this arithmetic progression is 88, then q-2p is equal to .

Let a1,a2,a3, be a G.P. of increasing positive terms. If a1a5=28 and a2+a4=29, then a6 is equal to:

1. 812
2. 784
3. 628
4. 526

If r=1nTr= (2n-1)(2n+1)(2n+3)(2n+5)64, then n r=1n ( 1Tr ) is equal to :

1. 23
2. 13
3. 1
4. 0

Suppose that the number of terms in an A.P. is 2k, k . If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to:

1. 8
2. 6
3. 4
4. 5