Sequences and Series

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