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Suppose four distinct positive numbers a...

Suppose four distinct positive numbers `a_(1),a_(2),a_(3),a_(4)` are in G.P. Let `b_(1)=a_(1)+,a_(b)=b_(1)+a_(2),b_(3)=b_(2)+a_(3)andb_(4)=b_(3)+a_(4)`.
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.

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Suppose 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_(1)

Suppose 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) . Statement - 1. The numbers b_(1), b_(2), b_(3), b_(4) are neither in A.P. nor in G.P. and Statement - 2. The numbers b_(1), b_(2), b_(3), b_(4) are in H.P.

Knowledge Check

  • Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let /_\=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then

    A
    `/_\ "is independent of "a_(1),a_(2),a_(3),b_(1),b_(2),b_(3)`
    B
    `a_(1)-/_\,a_(2)-2/_\,a_(3)-3/_\` are in H.P.
    C
    `b_(1)+/_\,b_(2)+/_\^(2),b^(3)+/_\` are in H.P.
    D
    none of these

  • If a and b are distinct positive real numbers such that a, a_(1), a_(2), a_(3), a_(4), a_(5), b are in A.P. , a, b_(1), b_(2), b_(3), b_(4), b_(5), b are in G.P. and a, c_(1), c_(2), c_(3), c_(4), c_(5), b are in H.P., then the roots of a_(3)x^(2)+b_(3)x+c_(3)=0 are

    A
    real and distinct
    B
    real and equal
    C
    imaginary
    D
    none of these

  • Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let Delta=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then prove that

    A
    `Delta` is independent of `a_(1),a_(2),a_(3)`
    B
    `A_(1)-Delta,a_(2)-2Delta,a_(3)-3Delta` are in A.P.
    C
    `b_(1)+Delta,b_(2)+Delta^(2),b_(3)+Delta` are in H.P.
    D
    `Delta` is independent of `b_(1),b_(2),b_(3)`

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