Sequences and Series (Progressions)

1

Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be

  1. 164√3
  2. 188√3
  3. 248√3
  4. 192√3
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2

Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is:

  1. 3/6
  2. 3/2
  3. 5/2
  4. 1/6
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3

What is the sum of the following series?

- 64, - 66, - 68, …… , - 100

  1. - 1458
  2. - 1558
  3. - 1568
  4. - 1664
  5. None of the above
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4

If three positive real numbers x, y and z satisfy y - x = z - y  and xyz = 4, then what is the minimum possible value of y?

  1. 2(1/4)
  2. 2(2/3)
  3. 2(1/3)
  4. 2(3/4)
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5

If log3 2, log3 (2x - 5), log3 (2x - 7/2) are in arithmetic progression, then the value of x is equal to

  1. 2
  2. 3
  3. 4
  4. 5
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6

Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

  1. 741
  2. 780
  3. 800
  4. 820
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7

Consider the set S = {1, 2, 3,..., 1000}. How many arithmetic progressions can be formed from the elements of S that start with 1 and end with 1000 and have at least 3 elements?

  1. 3
  2. 4
  3. 6
  4. 7
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8

For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 7th and 6th terms of this sequence is 517, what is the 10th term of this sequence?

  1. 76
  2. 108
  3. 123
  4. 147
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9

Fourth term of an arithmetic progression is 8. What is the sum of the first 7 terms of the arithmetic progression?

  1. 7
  2. 35
  3. 56
  4. 64
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10

If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio

  1. 12 : 13
  2. 4 : 9
  3. 2 : 3
  4. 1/12 : 1/13
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11

The number of common terms in the two sequences 17, 21, 25,..., 417 and 16, 21, 26,..., 466 is

  1. 19
  2. 20
  3. 77
  4. 78
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12

The integers 1, 2,..., 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?

  1. 821
  2. 820
  3. 819
  4. 781
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13

If p, q and r are three unequal numbers such that p, q and r are in A.P., and p, r-q and q-p are in G.P., then p : q : r is equal to

  1. 1 : 2 : 3
  2. 2 : 3 : 4
  3. 3 : 2 : 1
  4. 1 : 3 : 4
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14

Seema has joined a new Company after the completion of her B.Tech from a reputed engineering college in Chennai. She saves 10% of her income in each of the first three months of her service and for every subsequent month, her savings are Rs. 50 more than the savings of the immediate previous month. If her joining income was Rs. 3000, her total savings from the start of the service will be Rs. 11400 in

  1. 6 months
  2. 12 months
  3. 18 months
  4. 24 months
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15

The sum of series, (–100) + (–95) + (–90) + …………+ 110 + 115 + 120, is:

  1. 0
  2. 220
  3. 340
  4. 450
  5. None of the above
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16

An infinite geometric progression a1, a2, a3,... has the property that an = 3(an+1 + an+2 +…) for every n ≥ 1. If the sum a1 + a2 + a3 + … = 32, then a5 is

  1. 1/32
  2. 2/32
  3. 3/32
  4. 4/32
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17

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. 26.5
  2. 28
  3. 29.5
  4. 31
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18

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

  1. 4/3
  2. 1
  3. 7/4
  4. 8/5
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