Sequence and series session-3

Sequence and Series Session-3

Problems from previous year papers have been discussed in the session. 

A progression is called a harmonic progression (H.P.) if the reciprocals of its terms are in A.P.

Standard form: 1/a, 1/(a +d), 1/(a +2d) .........  1, 1/2, 1/3, ......... are in harmonic progression and 1, 2, 3..... are in arithmetic progression.

Harmonic Mean:

If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.

Also, if a, H, b are in H.P., then H is called harmonic mean between a and b.

1. The first term of an A.P. of consecutive integer is p² + 1. The sum of (2p + 1) terms of this series can be expressed as

(a) (p + 1)²                                    (b) (2p +1) (p+1)²

(c) (p+1)³                                      (d) p³ + (p+1)³

2. If S is the sum to infinity of G.P. whose first term is 'a', then the sum of the first n terms is

(a)                                   (b)    

(c)                         (d) None of these

3. If x > 0, and log₂x + log₂() + log₂() + log₂() + log₂() + … = 4, then x equals

(a) 2                    (b) 3                    (c) 4                   (d) 5

4. If A, G & H are respectively the A.M., G.M. & H. M. of three positive numbers a, b & c, then the equation whose roots are a, b & c is given by :

(a) x³ – 3Ax² + 3 G³x – G³ = 0

(b) x³ – 3 Ax² + 3 (G³/H) x – G³ = 0

(c) x³ + 3 Ax² + 3 (G³/H) x – G³  = 0

(d) x³ – 3 Ax² – 3 (G³/H) x + G³ = 0

5. Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are :

(a) not in A.P./G.P./H.P.          (b) in A.P.

(c) in G.P.                                     (d) in H.P 

6. If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a+b)(c+d) satisfies the relation :

(a) 0 < M < 1  (b) 1 < M < 2    (c) 2 < M < 3    (d) 3 < M < 4

7. If one A.M. A and two G.M.’s p and q be inserted between any two numbers, then the value of p³ + q³ is

(a)  2pq/A         (b) 2 A pq        (c) 2A p²q²        (d) none of these

8. If a₁,a₂, ...., a_n are positive numbers such that a₁. a_2. .... a_n = 1, then their sum is

(a) a positive integer                  (b) divisible by n

(c) never less than n                   (d) none of these

9. log ₂ x + log₂ y > or = 6 what is the smallest possible value of x + y

(a) 8                    (b) 16                 (c) 31                  (d) 32

10. An infinite G.P. has first term as x and sum upto infinity as 5. Then the range of values of ‘x’ is :

(a) x < – 10       (b) x > 10           (c) 0 < x < 10    (d) – 10 < x < 10

 Answers