Harmonic Progression

Q. If the Harmonic mean of the numbers $2,3,4$ is $H$ then $13H=$

Q. The first negetive term in the sequence of $56,55\frac{1}{5},54\frac{2}{5},\cdots$

1. ${72}^{th}$
2. ${73}^{th}$
3. ${70}^{th}$
4. ${71}^{th}$

Q.

Let $a_1,a_2,a_3,.............$ be in harmonic progression with $a_1=5and{a}_{20}=25.$ The least positive integer n for which $a_n<0$ is

1. 22

2. 23

3. 24

4. 25

Q. If $a_1,a_2,a_3$ are in A.P., $a_2,a_3,a_4$ are in G.P., and $a_3,a_4,a_5$ are in H.P., then $\ln a_1,\ln a_3,\ln a_5$ are in

1. G.P.
2. A.P.
3. H.P.
4. None of these

Q. If the 8th term of an H.P. is 1/2 and the 14th term is 1/3, then the 20th term is

1. $\frac{1}{5}$
2. $\frac{1}{4}$
3. $-\frac{1}{2}$
4. $\frac{1}{6}$

Q.

Let $a_1,a_2,a_3,.............$ be in harmonic progression with $a_1=5and{a}_{20}=25.$ The least positive integer n for which $a_n<0$ is

1. 23

2. 24

3. 22

4. 25

Q. Let the $n^{th}$ terms of an A.P., G.P. and H.P. be $a,b,c$ respectively. If the first and the $(2n-1{)}^{th}$ terms of the A.P., G.P. and H.P. are equal, then which of the following is/are correct?

1. $a\le b\le c$
2. $a\ge b\ge c$
3. $bc-a^2=0$
4. $ac-b^2=0$

Q. If $x_1,x_2,\cdots ,x_n$ and $\frac{1}{h_1},\frac{1}{h_2},\cdots ,\frac{1}{h_n}$ are two $A.P.$s such that $x_3=h_2=8$ and $x_8=h_7=20,$ then $x_5\cdot h_{10}$ equals

1. $2560$
2. $2650$
3. $3200$
4. $1600$

Q.

If the roots of $a(b-c)x^2+b(c-a)x+c(a-b)$ = 0 be equal then a, b, c are in

1. A.P

2. G.P

3. H.P

4. None of these

Q.

If a, b, c are three distinct positive real numbers which are in H.P., then $\frac{3a+2b}{2a-b}+\frac{3c+2b}{2c-b}$ is

1. Greater than or equal to 10

2. Less than or equal to 10

3. Only equal to 10

4. None of these

Q.

If $lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ be in H.P., then a, b, c are in

1. A.P

2. H.P

3. G.P

4. None of these

Q. If the roots of $10x^3-c{x}^2-54x-27=0$ are in H.P., then the value of $c$ is

Q.

If a, b, c, d are in H.P., then ab + bc + cd is equal to

1. 3ad

2. (a + b)(c + d)

3. 3ac

4. None of these

Q. If $a_1,a_2,a_3,\dots \dots$ are in a harmonic progression with $a_1=5$ and $a_{20}=25$. Then, the least positive integer n for which $a_n<0$, is

1. 23
2. 24
3. 22
4. 25

Q.

If a, b, c are in H.P then $4^{-a},4^{-b},4^{-c}$ are in

1. A.P

2. G.P

3. H.P

4. none of these

Q.

If a, b, c be in A.P. and b, c, d be in H.P., then

1. ab = cd

2. ad = bc

3. bc = ad

4. abcd = 1

Q. If $x^2+9y^2+25z^2=xyz(\frac{15}{x}+\frac{5}{y}+\frac{3}{z})$ then,

1. $x,y$ and $z$ are in $H.P.$
2. $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $A.P.$
3. $x,y$ and $z$ are in $G.P.$
4. $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $G.P.$

Q.

If a, b, c, are in H.P. then

1. $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$

2. All are correct

3. $\frac{a-b}{b-c}=\frac{a}{c}$

4. $\frac{b+a}{b-a}+\frac{b+c}{b-c}=2$

Q. If $x,y,z$ are real and $9x^2+16y^2+25z^2-12xy-20yz-15zx=0,$ then $x,y,z$ are in

1. A.P.
2. G.P.
3. H.P.
4. None of the above