Questions

1. Question 1

   Find the 9th term of the following harmonic sequences:

   (i)

   $\dfrac{1}{3},\ \dfrac{1}{5},\ \dfrac{1}{7},\ \ldots$

   (ii)

   $\dfrac{-1}{5},\ \dfrac{-1}{3},\ -1,\ \ldots$

   SolutionTheory
2. Question 2

   Insert five harmonic means between the following given numbers:

   (i)

   $\dfrac{-2}{5}$ and $\dfrac{2}{13}$

   (ii)

   $\dfrac{1}{4}$ and $\dfrac{1}{24}$

   SolutionTheory
3. Question 3

   The first term of an H.P. is $-\dfrac{1}{3}$ and the fifth term is $\dfrac{1}{5}$. Find its 9th term.

   SolutionTheory
4. Question 4

   If 5 is the harmonic mean between 2 and $b$, find $b$.

   SolutionTheory
5. Question 5

   If the numbers $\dfrac{1}{k},\ \dfrac{1}{2k+1}$ and $\dfrac{1}{4k-1}$ are in harmonic sequence, find $k$.

   SolutionTheory
6. Question 6

   Find $n$ so that $\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be H.M. between $a$ and $b$.

   SolutionTheory
7. Question 7

   If $a^2,\ b^2$ and $c^2$ are in A.P., show that $a+b,\ c+a$ and $b+c$ are in H.P.

   SolutionTheory
8. Question 8

   If the H.M. and A.M. between two numbers are 4 and $\dfrac{9}{2}$ respectively, find the numbers.

   SolutionTheory
9. Question 9

   If the (positive) G.M. and H.M. between two numbers are 4 and $\dfrac{16}{5}$, find the numbers.

   SolutionTheory
10. Question 10

    If $\dfrac{b+c-a}{a},\ \dfrac{c+a-b}{b},\ \dfrac{a+b-c}{c}$ are in A.P., show that $a,\ b,\ c$ are in H.P.

    SolutionTheory
11. Question 11

    If $a,\ b,\ c,\ d$ are in H.P., show that $3(a-b)(c-d)=(b-c)(a-d)$.

    SolutionTheory
12. Question 12

    If between any two numbers there are inserted two A.Ms. $A_1,\ A_2$, two G.Ms. $G_1,\ G_2$ and two H.Ms. $H_1,\ H_2$; show that $\dfrac{A_1+A_2}{G_1 G_2} = \dfrac{H_1+H_2}{H_1 H_2}$.

    SolutionTheory
13. Question 13

    The H.M. of two numbers is 4. The A.M., $A$ and the G.M., $G$ satisfy the relation $2A + G^2 = 27$. Find the numbers.

    SolutionTheory
14. Question 14

    First three of the four numbers $a,\ b,\ c,\ d$ are in A.P., and the next three are in H.P., show that $ad = bc$.

    SolutionTheory
15. Question 15

    If $a,\ b,\ c$ are in G.P., show that $\log_a x,\ \log_b x,\ \log_c x$ are in H.P.

    SolutionTheory
16. Question 16

    If $a,\ b,\ c$ are in H.P., show that

    (i)

    $\dfrac{a-b}{b-c} = \dfrac{a}{c}$

    (ii)

    $(a-c)^2 = (a+c)(a-2b+c)$

    SolutionTheory
17. Question 17

    If $2+x,\ 5+x$ and $9+x$ are in H.P., find the value of $x$.

    SolutionTheory
18. Question 18

    If the roots of the equation $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ are equal, prove that $a,\ b,\ c$ are in H.P.

    SolutionTheory