Sequence and series session-2

Sequence and Series Session-2

Basic concepts and problems on arithmetic, geometric and harmonic progression have been discussed in the video lecture. Useful for the students appearing for competitive exams.

Try these problems :

1. If the sides of a right angled triangle form an A.P. then the sines of the acute angle is

(a) 3/5, 4/5     (b) 3/4, 3/5       (c) 2/5, 3/5      (d) none of these

2. If the length of sides of right angled triangle are in AP, then their ratio is

(a) 3 : 4 : 5      (b) 2 : 3 : 4         (c) 1 : 2 : 3         (d) none of these

3. If the ratio of sums to n terms of two A.,P.’s is (5n + 3) : (3n+ 4) then the ratio of their 17^th terms is

(a) 172 : 99         (b) 168 : 103  (c) 175 : 99        (d) 171 : 103

4. If the sum of an infinite G.P. and the sum of the squares of its terms are both equal to 5, then the first term is

(a) 5/3               (b) 2/3               (c) 7/3                (d) none of these

5. The sum of two numbers is 2 1/6 and even numbers of A.M’s are inserted between them. If the sum of these means exceeds their number by 1, then the number of means is

(a) 11                   (b) 12                 (c) 13                  (d) 14

6. The first term of an infinite G.P. is 1 and any term is equal to the sum of all the succeeding terms. The common ratio of the G.P. is

(a) 1/3                (b) 1/2               (c) 1/6                (d) 1/4

7. If a₁, a₂, a₃, ... are in A.,P. such that a₁ + a₅ + a₁₀ + a₁₅ + a₂₀ + a₂₄ = 225, then a₁ + a₂ + ….. + a₂₃ + a₂₄ is equal to

(a) 909               (b) 75                  (c) 750                (d) 900

8. If x belongs to R, the numbers 5^1+x + 5^1–x, a/2, 25^x + 25^–x form an A.P. then ‘a’ must lie in the interval

(a) [1,5]              (b) [2,5]             (c) [5,12]             (d) [12, inf)

9. Suppose a, b, c are in A.P. & |a|, |b|, |c| < 1. If x = 1 + a + a² + ... to inf, y = 1 + b + b² + ... to ¥ & z = 1 + c + c² + ... to inf then x, y, z are in :

(a) A.P.               (b) G.P.              (c) H.P.            (d) none

10. a, b be the roots of the equation x² – 3x + a = 0 and g, d the roots of x² – 12x + b = 0 and numbers a, b, g, d (in this order) form an increasing G.P., then

(a) a = 3, b = 12                            (b) a = 12, b = 3

(c) a = 2, b = 32                       (d) a = 4, b = 16

11. Let a₁, a₂, a₃, .. be terms of an A.P. if (a₁ + a₂ + … + a_p)/(a₁ + a₂ + … + a_q) = p²/q², then a₆/a₂₁ equals

(a) 41/11             (b) 7/2                (c) 2/7                (d) 11/41

12. If a₁, a₂, … , a_n are in H.P. then the expression a₁a₂ + a₂a₃ + … + a_n–1a_n is equal to

(a) (n – 1)a₁a_n                          (b) n (a₁ – a_n)

(c) (n – 1) (a₁ – a_n)                       (d) n a₁ a_n

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

(a) x² + 18x – 16 = 0                    (b) x² – 18x + 16 = 0

(c) x² + 18x + 16 = 0                    (d) x² – 18x – 16 = 0

14. 1³ – 2³ + 3³ – 4³ + …. 9³ =                                                          

(a) 425              (b) –425            (c) 475                (d) –475

15. If x_i > 0, i = 1, 2, …, 50 and x₁ + x₂ + … + x₅₀ = 40, then the minimum value of  equals to

(a) 50                (b) (50)²             (c) (50)³             (d) (50)⁴

16. The sum of the products of every pair of the first n natural numbers is

(a)               (b)

(c)                (d) None of these

17. If  up to , then  equals to

(a) π²/6              (b) π²/16            (c) π²/8             (d) None of these

 Answers