Sequences and Series

If ( 1 + 1 + 1 + 2 + + 1 + 1012 ) - ( 12 1 + 14 3 + 16 5 + + 12024 2023 ) = 12024, then is equal to .

Let the positive integers be written in the form : [figure] If the kth row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is .

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 703 and the product of the third and fifth terms is 49. Then the sum of the 4th, 6th and 8th terms is equal to:

1. 78
2. 96
3. 91
4. 84

An arithmetic progression is written in the following way [figure] The sum of all the terms of the 10th row is .

Let the first term of a series be T1 = 6 and its rth term Tr = 3Tr-1 + 6r, r = 2, 3, , n. If the sum of the first n terms of this series is 15 (n2 - 12n + 39 ) (4 6n - 5 3n + 1 ), then n is equal to .

n (12 - 1)(n - 1) + (22 - 2)(n - 2) + + ((n - 1)2 - (n - 1)) 1(13 + 23 + + n3) - (12 + 22 + + n2) is equal to :

1. 23
2. 12
3. 34
4. 13

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is the sum of areas of all the triangles formed in this process, then :

1. P2 = 72 3Q
2. P2 = 36 3Q
3. P = 36 3Q2
4. P2 = 6 3Q

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to:

1. 125
2. 160
3. 150
4. 180

If S(x) = (1 + x) + 2(1 + x)2 + 3(1 + x)3 + + 60(1 + x)60, x 0, and (60)2 S(60) = a(b)b + b, where a, b N, then (a + b) equal to .

If 1 1 + 2 + 1 2 + 3 + + 1 99 + 100 = m and 11 2 + 12 3 + + 199 100 = n, then the point (m, n) lies on the line

1. 11(x - 1) - 100y = 0
2. 11x - 100y = 0
3. 11(x - 1) - 100(y - 2) = 0
4. 11(x - 2) - 100(y - 1) = 0