Home
Class 12
MATHS
If a,b,c,d be four distinct positive qua...

If `a,b,c,d` be four distinct positive quantities in GP,then
(a) `a+dgtb+c`
(b) ` c^(-1)d^(-1)+a^(-1)b^(-1)gt2(b^(-1)d^(-1)+a^(-1)c^(-1)-a^(-1)d^(-1))`

Text Solution

Verified by Experts

`therefore a,b,c,d` are in GP.
(a) Applying AMgtGM
For first three members,
`(a+c)/(2)gtb`
`implies a+cgt2b " " "….(v)"`
For last three members, `(b+d)/(2)gtc`
`implies b+dgt2c " " "…..(vi)"`
From Eqs. (v) and (vi), we get
`a+c+b+dgt2b+2c " or " a+dgtb+c`
(b) Applying GMgt HM
For first three members, `bgt(2ac)/(a+c)`
`implies ab+bcgt 2ac " " "....(vii)"`
For last three members, `cgt(2bd)/(b+d)`
`implies bc+cdgt2bd " " "....(viii)"`
From Eqs. (vii) and (viii), we get
`ab+bc+bc+cdgt2ac+2bd`
or `ab+cdgt 2(ac+bd-bc)`
Dividing in each term by `abcd`, we get
` c^(-1)d^(-1)+a^(-1)b^(-1)gt2(b^(-1)d^(-1)+a^(-1)c^(-1)-a^(-1)d^(-1))`
Promotional Banner

Topper's Solved these Questions

  • SEQUENCES AND SERIES

    ARIHANT MATHS ENGLISH|Exercise Exercise For Session 1|5 Videos
  • SEQUENCES AND SERIES

    ARIHANT MATHS ENGLISH|Exercise Exercise For Session 2|11 Videos
  • PROPERTIES AND SOLUTION OF TRIANGLES

    ARIHANT MATHS ENGLISH|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|21 Videos
  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|12 Videos

Similar Questions

Explore conceptually related problems

If a,b,c,d be four disinct positive quantities in HP, then (a) a+dgtb+c (b) adgt bc

If a ,b ,c ,d are four distinct positive numbers in G.P. then show that a+d > b+c dot

If A and B are invertible matrices, which of the following statement is not correct. a d j\ A=|A|A^(-1) (b) det(A^(-1))=(detA)^(-1) (c) (A+B)^(-1)=A^(-1)+B^(-1) (d) (A B)^(-1)=B^(-1)A^(-1)

If a, b, c are distinct positive real numbers such that a+(1)/(b)=4,b+(1)/( c )=1,c+(1)/(d)=4 and d+(1)/(a)=1 , then

For non-singular square matrix A ,\ B\ a n d\ C of the same order then, (A B^(-1)C)^(-1)= (a) A^(-1)B C^(-1) (b) C^(-1)B^(-1)A^(-1) (c) C B A^(-1) (d) C^(-1)\ B\ A^(-1)

If a,b,c,d are conseentive binomial coefficients of (1+x)^n then (a+b)/(a), (b+c)/(b), (c+d)/(c ) are is

If a,b,c and d are four positive real numbers such that abcd=1 , what is the minimum value of (1+a)(1+b)(1+c)(1+d) .

If a, b, c , d are in G.P. , then shown that (i) (a + b)^(2) , (b +c)^(2), (c + d)^(2) are in G.P. (ii) (1)/(a^(2) + b^(2)), (1)/(b^(2) +c^(2)), (1)/(c^(2) + d^(2)) are in G.P.

Two different non-parallel lines cut the circle |z| = r at points a, b, c and d , respectively. Prove that these lines meet at the point z given by (a^-1+b^-1-c^-1-d^-1)/(a^-1b^-1-c^-1d^-1)

If a ,b ,c ,d are in G.P., then prove that (a^3+b^3)^(-1),(b^3+c^3)^(-1),(c^3+d^3)^(-1) are also in G.P.

ARIHANT MATHS ENGLISH-SEQUENCES AND SERIES-Exercise (Questions Asked In Previous 13 Years Exam)
  1. If a,b,c,d be four distinct positive quantities in GP,then (a) a+dgt...

    Text Solution

    |

  2. Let a,b,c be in A.P. and |a|lt1,|b|lt1|c|lt1.ifx=1+a+a^(2)+ . . . ."to...

    Text Solution

    |

  3. about to only mathematics

    Text Solution

    |

  4. If a1, a2, a3, be terms of an A.P. and (a1+a2+.....+ap)/(a1+a2+.....+...

    Text Solution

    |

  5. If a1, a2, a3,.....an are in H.P. and a1 a2+a2 a3+a3 a4+.......a(n-1...

    Text Solution

    |

  6. Let V(r ) denotes the sum of the first r terms of an arithmetic progre...

    Text Solution

    |

  7. Let Vr denote the sum of the first r terms of an arithmetic progressio...

    Text Solution

    |

  8. Let V(r) denote the sum of the first r terms of an arithmetic progress...

    Text Solution

    |

  9. LetA(1),G(1),H(1) denote the arithmetic, geometric and harmonic means ...

    Text Solution

    |

  10. Let A1 , G1, H1denote the arithmetic, geometric and harmonic means re...

    Text Solution

    |

  11. LetA(1),G(1),H(1) denote the arithmetic, geometric and harmonic means ...

    Text Solution

    |

  12. In a G.P of positive terms if any term is equal to the sum of the next...

    Text Solution

    |

  13. Suppose four distinct positive numbers a(1),a(2),a(3),a(4) are in G.P....

    Text Solution

    |

  14. The first two terms of a geometric progression add up to 12. The sum o...

    Text Solution

    |

  15. If the sum of first n terms of an A.P. is cn^(2) then the sum of squar...

    Text Solution

    |

  16. The sum to infinity of the series 1+2/3+6/3^2+14/3^4+...is

    Text Solution

    |

  17. Let Sk,k=1, 2, …. 100 denote the sum of the infinite geometric series ...

    Text Solution

    |

  18. Let a1, a2, a3, ,a(11) be real numbers satisfying a1=15 , 27-2a2>0 a ...

    Text Solution

    |

  19. A person is to count 4500 currency notes. Let an denote the number of ...

    Text Solution

    |

  20. The minimum value of the sum of real numbers a^-5, a^-4, 3a^-3, 1,a^8 ...

    Text Solution

    |

  21. A man saves ₹ 200 in each of the first three months of his servies.In ...

    Text Solution

    |