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(a) Consider an infinite geometric series with first term a and common ratio r. If the sum is 4 and the second term is 3/4, then [JEE 2000, (scr.), 1 + 1]
(A) a = , r = (B) a = 2, r = (C) a = , r = (D) a = 3, r =
(b) If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisifes the relation :
(A) 0 £ M £ 1 (B) 1 £ M £ 2 (C) 2 £ M £ 3 (D) 3 £ M £ 4
(c) The fourth power of the common difference of an arithmetic progression with integer entries added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer. [JEE 2000, (Mains), 4]
Given that a, g are roots of the equation, Ax^{2}_4x+1 =0 and b, d the roots of the equation, Bx^{2} _ 6x + 1 = 0, find values of A and B, such that a, b, g & d are in H.P. [REE 2000, 5]
The sum of roots of the equation ax^{2} + bx + c = 0 is equal to the sum of squares of their reciprocals. Find whether bc^{2}, ca^{2} and ab^{2} in A.P., G.P. or H.P. ? [ree 2001, 3]
Solve the following equations for x and y log_{2}x + log_{4}x + log_{16}x +..............
........... = y = 4log_{4} x [ree 2001, 5]
a) Le a, b be the roots of x^{2} _ x + p = 0 and g, d the roots of x^{2} _ 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 a_{1}, a_{2},..... 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 a_{1}, a_{2},..........., a_{n}. Find an expression for the G.M. of G_{1}, G_{2},............, G_{n} in terms of A_{1}, A_{2},...........,A_{n}, H_{1}, H_{2},........,H_{n}. [JEE 2001 (Mains), 5]
(a) Suppose a, b, c are in A.P. and a^{2}, b^{2}, c^{2} 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, A_{1}, A_{2},b are in A.P. ; a, G_{1}, G_{2}, b are in G.P. and a, H_{1}, H_{2}, 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 ax^{2} + bx + c = 0, If D = b^{2} _ 4ac and a + b, a^{2} + b^{2}, a^{3}+ b^{3}, are in G.P. where a, b are the roots of ax^{2} + 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 (2^{n+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 n_{0} such that B_{n} > A_{n} n > n_{0}. [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 V_{1} + V_{2} + ....... + V_{n} is
(A) n(n+1)(3n^{2} _n+1) (B) n(n+1)(3n^{2}+n+2) (C) n (2n^{2}_n+1) (D) (2n^{3}_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) Q_{1}, Q_{2}, Q_{3},.... are in A.P., with common difference 5
(B) Q_{1}, Q_{2}, Q_{3},.... are in A.P., with common difference 6
(C) Q_{1}, Q_{2}, Q_{3},.... are in A.P., with common difference 11
(D) Q_{1 }= Q_{2 }= Q_{3 }=.... [JEE 2007, 4 + 4 + 4]
Let A_{1}, G_{1}, H_{1} 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) G_{1} > G_{2} > G_{3} > ........ (B) G_{1} < G_{2} < G_{3} < ........
(C) G_{1} = G_{2} = G_{3} = ........ (D) G_{1} < G_{3} < G_{5} < ...... and G_{2} > G_{4} > G_{6} > ......
(b) Which one of the following statement is correct ?
(A) A_{1} > A_{2} > A_{3} > ........ (B) A_{1} < A_{2} < A_{3} < ........
(C) A_{1} > A_{3} > A_{5} > .....and A_{2} < A_{4} < A_{6} < ........ (D) A_{1} < A_{3} < A_{5} < .....and A_{2} > A_{4} > A_{6} > ........
(c) Which one of the following statement is correct ? [JEE 2007, 4 + 4 + 4]
(A) H_{1} > H_{2} > H_{3} > ........ (B) H_{1} < H_{2} < H_{3} < ........
(C) H_{1} > H_{3} > H_{5} > .....and H_{2} < H_{4} < H_{6} < ........ (D) H_{1} < H_{3} < H_{5} < .....and H_{2} > H_{4} > H_{6} > ........
(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 a_{1}, a_{2}, a_{3}, a_{4} are in G.P. Let b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3} and b_{4} = b_{3} + a_{4}. [JEE 2008, 3]
Statement_1 : The numbers b_{1}, b_{2}, b_{3}, b_{4} are neither in A.P. nor in G.P.
Statement_2 : The numbers b_{1}, b_{2}, b_{3}, b_{4} 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 cn^{2}, then the sum of squares of these n terms is [JEE 2009, 3]
Sn = cn^{2}
S(n−1) = c(n−1)^{2}
= cn^{2}+c−2cn
T_{n} = 2cn − c
T_{n}^{2} = (2cn−c)^{2}
= 4c^{2}n^{2} + c^{2} − 4c^{2} n
Required sum = ∑Tn^{2} = 4c^{2} ∑n^{2} +nc^{2} −4c^{2} ∑n
{4c^{2}n(n+1)(2n+1)}/6 + nc^{2}−2c^{2}n(n+1)
= (2c^{2}n(n+1)(2n+1)+3nc^{2}−6c^{2}n(n+1))/3
= {nc^{2}[4n^{2}+6n+2+3−6n−6]}/3
= [nc^{2}(4n^{2}−1)]/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]
Infinite Sum of GP S = a/(1−r) = (k−1)/[k!(1−1/k)] = 1/(k−1)!
(100)^{2}/100! + ∑k=(1 to 100)(k2−3k+1)Sk
(100)^{2}/100! + ∑k=(1 to 100) [(k1)^2 k[(1/(k1)!]
(100)^{2}/100! + ∑k=(2 to 100)[(k1)/(k2)!  k/(k1)!
(100)^{2}/100! 2/1!  1/0 + 2/1!  3/2! + 3/2! +.....+ 99/98!  100/99!
= (100)^{2}/100! + 2  1 + 2  dfrac{100 * 100}{99! * 100}
= 3
Let a_{1}, a_{2}, a_{3}, ....., a_{11} be real numbers satisfying
a_{1} = 15, 27 _ 2a_{2} > 0 and a_{k} = 2a_{k1} _ a_{k2} 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, a^{8} and a^{10} with a > 0 is [JEE 2011]
Let a_{1}, a_{2}, a_{3}, ..., a_{100} be an arithmetic progression with a_{1} = 3 and S_{p} = , 1 £ p £ 100. For any integer n with 1 £ n £ 20. let m=5n. Ifdoes not depend on n, then a_{2} is [JEE 2011]
Let a_{1}, a_{2}, a_{3}, ..... be in harmonic progression with a_{1} = 5 and a_{20} = 25. The least positive integer n for which a_{n} < 0 is [JEE 2012]
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