nth Term of HP

Q. If $S_1,S_2,S_3$ denote the sum of first $11$ terms of three arithmetic progressions whose first terms are unity and their common differences are in harmonic progressions, then the value of $\left(\frac{2S_3{S}_1-S_1{S}_2-S_2{S}_3}{S_1-2S_2+S_3}\right)$ is

Q. If $a,b,c(a<b<c)$ are in H.P. such that $a+b+c=37$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{4}$, then which of the following is/are correct?

1. $a,b,c-1$ are in A.P.
2. $a-4,b,c+9$ are in G.P.
3. A.M. of $a,b$ is $15$
4. $\frac{a}{2},\frac{b}{3},\frac{c}{5}$ are sides of a right angled triangle

Q.

If the seventh term of an A.P. is $\frac{1}{9}$ and its ninth term is $\frac{1}{7}$, find its $63rd$ term.

Q. If $a,b,c(a<b<c)$ are in H.P. such that $a+b+c=37$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{4}$, then which of the following is/are correct?

1. $a,b,c-1$ are in A.P.
2. $a-4,b,c+9$ are in G.P.
3. A.M. of $a,b$ is $15$
4. $\frac{a}{2},\frac{b}{3},\frac{c}{5}$ are sides of a right angled triangle

Q.

If the $7^{th}$ term of a harmonic progression is 8 and the $8^{th}$ term is 7, then its ${15}^{th}$ term is

1. 16

2. 14

3. $\frac{27}{14}$

4. $\frac{56}{15}$

Q.

The first term of a H P whose second term is $\frac{5}{2}$ and third term is $\frac{10}{3}$, is___

Q. Given that $\alpha ,\beta ,a,b$ are in A.P., $\alpha ,\beta ,c,d$ are in G.P. and $\alpha ,\beta ,e,f$ are in H.P. If $b,d,f$ are in G.P., then the value of $\frac{2({\alpha }^6-{\beta }^6)}{\alpha \beta ({\alpha }^4-{\beta }^4)},0<\alpha <\beta$ is

Q. If the $7^{\text{th}}$ and ${12}^{\text{th}}$ terms of a H.P. are $\frac{1}{10}$ and $\frac{1}{25}$ respectively, then the ${20}^{\text{th}}$ term of H.P. is

1. $\frac{1}{37}$
2. $\frac{1}{42}$
3. $\frac{1}{45}$
4. $\frac{1}{49}$

Q.

If the $m^{th}$ term of a H.P. be n and $n^{th}$ be m, then the $r^{th}$ term will be

1. $\frac{r}{mn}$

2. $\frac{mn}{r+1}$

3. $\frac{mn}{r}$

4. $\frac{mn}{r-1}$

Q.

If sixth term of a H.P. is $\frac{1}{61}$ and its tenth term is $\frac{1}{105}$, then first term of that H.P. is

1. $\frac{1}{28}$

2. $\frac{1}{39}$

3. $\frac{1}{6}$

4. $\frac{1}{17}$

Q.

The 4 term of a H.P is $\frac{3}{5}$ and 8th term is $\frac{1}{3}$, then its 6th term is

1. $\frac{1}{6}$

2. $\frac{3}{7}$

3. $\frac{1}{7}$

4. $\frac{3}{5}$