Sequences and Series

1

For any three positive real numbers a, b and c, 9(25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c). Then

  1. b, c and a are in G.P.
  2. b, c and a are in A.P.
  3. a, b and c are in A.P.
  4. a, b and c are in G.P.
2

If the mean deviation about the median of the numbers a, 2a,..., 50a is 50, then |a| equals

  1. 2
  2. 3
  3. 4
  4. 5
3

If the mean deviation of the numbers 1, 1 + d, 1 + 2d,... , 1 + 100d from their mean is 255, then the d is equal to

  1. 10
  2. 10.1
  3. 20
  4. 20.2
4

Let f(x) be a polynomial function of second degree. If f(1) = f(–1) and a, b, c are in A.P., then f'(a), f'(b) and f'(c) are in

  1. G.P.
  2. A.P.
  3. A.P. - G.P.
  4. H.P.
5

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is

  1. 31
  2. 29.5
  3. 28
  4. 26.5
6

The sum to infinity of the series 1 + 2/3 + 6/32 + 10/33 +... is

  1. 2
  2. 3
  3. 4
  4. 6
7

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

  1. √2 + √3
  2. 2 + √3
  3. 2 - √3
  4. 3 + √2
8

If 100 times the 100th term of an AP with non zero common difference equals the 50 times its 50th term, then the 150th term of this AP is

  1. 0
  2. 150
  3. -150
  4. 150 times its 50th term
9

Fifth term of a GP is 2, then the product of its 9 terms is

  1. 128
  2. 256
  3. 512
  4. 1024
10

Sum of infinite number of terms in GP is 20 and sum of their square is 100. The common ratio of GP is

  1. 1/5
  2. 3/5
  3. 8/5
  4. 5
11

If the system of linear equations x + 2ay + az = 0, x + 3by + bz = 0, x + 4cy + cz = 0 has a non-­zero solution, then a, b, c

  1. satisfy a + 2b + 3c
  2. are in G.P
  3. are in A.P
  4. are in H.P
12

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

  1. x2 + 18x + 16 = 0
  2. x2 - 18x - 16 = 0
  3. x2 - 18x + 16 = 0
  4. x2 + 18x - 16 = 0
13

Let a and b be roots of the equation px2 + qx + r, p ≠ 0. If p, q, r are in A.P. and 1/a + 1/b = 4, then the value of |a - b| is

  1. 2√17 / 9
  2. √61 / 9
  3. √34 / 9
  4. 2√13 / 9