Test: Previous Year Questions- Sequences & Series- 2 - Commerce MCQ

Solve the following equations for x and y log₂x + log₄x + log₁₆x +..............

........... = y  = 4log₄ x     [ree 2001, 5]

a) Le a, b be the roots of x² _ x + p = 0 and g, d the roots of x² _ 4x + q = 0. If a, b, g, d are in G.P., then the integral values of p and q respectively, are

(A) _2, _32 (B) _2, 3 (C) _6, 3 (D) _6. _ 32

(b) If the sum of the first 2n terms of the A.P. 2, 5, 8,... ..........is equal to the sum of the first n terms of the A.P. 57, 59, 61,......., then n equals

(A) 10 (B) 12 (C) 11 (D) 13

(c) Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are [Jee 2001, (Scr.) 1 + 1 + 1]

(A) NOT in A.P./G.P./ H.P. (B) in A.P. (C) in G.P. (D) in H.P.

(d) Let a₁, a₂,..... be positive real numbers in G.P. For each n, let A_n, G_n, H_n be respectively, the arithmetic mean, geometric mean, and harmonic mean of a₁, a₂,..........., a_n. Find an expression for the G.M. of G₁, G₂,............, G_n in terms of A₁, A₂,...........,A_n, H₁, H₂,........,H_n.                 [JEE 2001 (Mains), 5]

 (a) Suppose a, b, c are in A.P. and a², b², c² are in G.P. if a < b < c and a + b + c = 3/2, then the value of a is [JEE 2002 (Scr.), 3]

(A)  (B)  (C)  (D) 

(b) Let a, b be positive real numbers. If a, A₁, A₂,b are in A.P. ; a, G₁, G₂, b are in G.P. and a, H₁, H₂, b are in H.P. show that== .                    [JEE 2002 (Mains), 5]

The first term of an infinite geometric progression is x and its sum is 5. Then         [JEE 2004 (Scr.)]

(a) In the quadratic equation ax² + bx + c = 0, If D = b² _ 4ac and a + b, a² + b², a³+ b³, are in G.P. where a, b are the roots of ax² + bx + c = 0, then [JEE 2005 (Scr.)]

(A) D ¹ 0 (B) bD = 0 (C) cD = 0 (D) D = 0

(b) If total number of runs scored in n matches is  (2ⁿ⁺¹ _ n _ 2) where n > 1, and the runs scored in the k^th match are given by k.2^n+1_k, where 1 £ k £ n. Find n.               [JEE 2005 (Mains), 2]

 If A_n =  _  + ............(_1)^{n _ 1} and B_n = 1 _ A_n, then find the minimum natural number n₀ such that B_n > A_n n > n₀.           [JEE 2006, 6]

 Let V_r denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r _ 1). Let T_r = V _{r + 1} _ V_r _ 2 and Q_r= T_{r + 1} _ T_r for r = 1, 2,.....

(a) The sum V₁ + V₂ + ....... + V_n is

(A) n(n+1)(3n² _n+1) (B) n(n+1)(3n²+n+2) (C) n (2n²_n+1) (D) (2n³_2n+3)

(b) T_r is always

(A) an odd number (B) an even number (C) a prime number (D) a composite number

(c) Which one of the following is a correct statement ?

(A) Q₁, Q₂, Q₃,.... are in A.P., with common difference 5

(B) Q₁, Q₂, Q₃,.... are in A.P., with common difference 6

(C) Q₁, Q₂, Q₃,.... are in A.P., with common difference 11

(D) Q₁ = Q₂ = Q₃ =....                                                        [JEE 2007, 4 + 4 + 4]

 Let A₁, G₁, H₁ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ³ 2, Let A_{n _ 1} and H_{n _ 1} have arithmetic, geometric and harmonic means as A_n, G_n, H_n respectively

(a) Which one of the following statements is correct ?

(A) G₁ > G₂ > G₃ > ........ (B) G₁ < G₂ < G₃ < ........

(C) G₁ = G₂ = G₃ = ........ (D) G₁ < G₃ < G₅ < ...... and G₂ > G₄ > G₆ > ......

(b) Which one of the following statement is correct ?

(A) A₁ > A₂ > A₃ > ........ (B) A₁ < A₂ < A₃ < ........

(C) A₁ > A₃ > A₅ > .....and A₂ < A₄ < A₆ < ........ (D) A₁ < A₃ < A₅ < .....and A₂ > A₄ > A₆ > ........

(c) Which one of the following statement is correct ?            [JEE 2007, 4 + 4 + 4]

(A) H₁ > H₂ > H₃ > ........ (B) H₁ < H₂ < H₃ < ........

(C) H₁ > H₃ > H₅ > .....and H₂ < H₄ < H₆ < ........ (D) H₁ < H₃ < H₅ < .....and H₂ > H₄ > H₆ > ........

(a) A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then     [JEE 2008, 4]

(A)  (B)  (C)  (D) 

(b) Supose four distinct positive numbers a₁, a₂, a₃, a₄ are in G.P. Let b₁ = a₁, b₂ = b₁ + a₂, b₃ = b₂ + a₃ and b₄ = b₃ + a₄. [JEE 2008, 3]

Statement_1 : The numbers b₁, b₂, b₃, b₄ are neither in A.P. nor in G.P.

Statement_2 : The numbers b₁, b₂, b₃, b₄ are in H.P.

(A) Statement (1) is true and statement (2) is true and statement (2) is correct explanation for (1)

(B) Statement (1) is true and statement (2) is true and statement (2) is NOT correct explanation for (1)

(C) Statement (1) is true but (2) is false

(D) Statement (1) is false but (2) is true

 If the sum of first n terms of an A.P. is cn², then the sum of squares of these n terms is    [JEE 2009, 3]

 Let S_k, K = 1, 2, ...., 100 denote the sum of the infinite geometric series whose first term is  and the common ratio is 1/k. Then the value of  is        [JEE 2010]

 Let a₁, a₂, a₃, ....., a₁₁ be real numbers satisfying

a₁ = 15, 27 _ 2a₂ > 0 and a_k = 2a_k-1 _ a_k-2 for k = 3, 4, ...., 11

If , then the value of  is equal to        [JEE 2010]

 The minimum value of the sum of real numbers a^_5, a^_4, 3a^_3, 1, a⁸ and a¹⁰ with a > 0 is    [JEE 2011]

 Let a₁, a₂, a₃, ..., a₁₀₀ be an arithmetic progression with a₁ = 3 and S_p = , 1 £ p £ 100. For any integer n with 1 £ n £ 20. let m=5n. Ifdoes not depend on n, then a₂ is     [JEE 2011]

Let a₁, a₂, a₃, ..... be in harmonic progression with a₁ = 5 and a₂₀ = 25. The least positive integer n for which a_n < 0 is         [JEE 2012]