Sequences and Series

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 .

Let the first three terms 2, p and q, with q 2, of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the 5th term of the G.P. is the nth term of the A.P., then n is equal to:

1. 151
2. 177
3. 163
4. 169

Let a=1+ 2C23!+ 3C24!+ 4C25!+ , b=1+ 1C0+1C11!+ 2C0+2C1+2C22! + 3C0+3C1+3C2+3C33!+ . Then 2ba2 is equal to .

The value of 1 22 + 2 32 + + 100 (101)212 2 + 22 3 + + 1002 101 is

1. 305301
2. 306305
3. 3231
4. 3130