Sequences and Series

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

Let a1, a2, a3, be in an arithmetic progression of positive terms. Let Ak = a12 - a22 + a32 - a42 + + a2k-12 - a2k2. If A3 = -153, A5 = -435 and a12 + a22 + a32 = 66, then a17 - A7 is equal to .

For x 0, the least value of K, for which 41 + x + 41 - x, K2, 16x + 16-x are three consecutive terms of an A.P., is equal to :

1. 10
2. 4
3. 8
4. 16

If 1 + 3 - 22 3 + 5 - 2 618 + 9 3 - 11 236 3 + 49 - 20 6180 + upto = 2 + ( ba + 1 ) e ( ab ), where a and b are integers with (a, b) = 1, then 11a + 18b is equal to .