nth Term of HP

Q.

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

1. $\frac{1}{37}$

2. $\frac{1}{41}$

3. $\frac{1}{45}$

4. $\frac{1}{49}$

Q. Prove by the Principle of Mathematical Induction: \(3^2n−1\) is divisible by 8, for all natural numbers n.

Q. If $a,b,c$ are in H.P. and $ab+bc+ca=15,$ then $ca=$

1. $5$
2. $7.5$
3. $3$
4. $4.5$

Q.

If $5^{th}$ term of a H.P. is $\frac{1}{45}$ and ${11}^{th}$ term is $\frac{1}{69}$, then its ${16}^{th}$ term will be

1. 

2. 

3. 

4. 

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. 14

2. 16

3. 

4. 

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 $\frac{{\beta }^6-{\alpha }^6}{\alpha \beta ({\beta }^4-{\alpha }^4)}$ is

1. $\frac{2}{3}$
2. $\frac{3}{2}$
3. $\frac{4}{3}$
4. $\frac{3}{4}$

Q.

Write the first four terms of each of the following sequences whose $n^{th}$term are $a_n=n^2-n+1$

Q. Prove by the Principle of Mathematical Induction: n^3−7n + 3 is divisible by 3, for all natural numbers n.

Q.

The value of ${\text{n}}_{{\mathrm{C}}_{\text{n}}}$ is

1. $\text{n}$

2. $0$

3. $1$

4. $\text{n}$!

Q.

Three numbers are in A.P. If the sum of these numbers is $27$ and the product is $648$, find the numbers.

Q. Find the sum of $n$ terms of the $A.P.,$ whose $k^{th}$ term is $5k+1.$

Q.

Let $f$ be a function defined by $f\left(x\right)=\left\{\begin{array}{ll}\frac{\tan x}{x},& x\ne 0\\ 1,& x=0\end{array}\right.$

Statement $\text{I}$: $x=0$ is a point of minima of $f$.

Statement $\text{II}$ : $f\left(0\right)=0$

1. Statement $\text{I}$ is correct, Statement $\text{II}$ is correct

2. Statement $\text{I}$ is correct, statement $\text{II}$ is correct; statement $\text{II}$ is the correct explanation for Statement $\text{I}$

3. Statement $\text{I}$ is correct, statement $\text{II}$ is correct; Statement $\text{II}$ is not a correct explanation for Statement $\text{I}$.

4. Statement $\text{I}$ is correct, statement $\text{II}$ is incorrect

Q. If three distinct numbers $a,b,c$ are in $H.P.$ and their squares are in $A.P.,$ then $a:b:c$ can be

1. $1-\sqrt{3}:-2:1+\sqrt{3}$
2. $1-\sqrt{3}:2:1+\sqrt{3}$
3. $2+\sqrt{3}:-2:2-\sqrt{3}$
4. $1+\sqrt{3}:-2:2-\sqrt{3}$

Q. If the roots of the equation $10x^3-n{x}^2-54x-27=0$ are in harmonic progression, then the value of ${}^{\prime }{n}^{\prime }$ is

Q. If log (x + z) + log(x – 2y + z) = 2 log(x – z), then x, y, z are in

1. A.P
2. H.P
3. A.GP
4. G.P

Q. If $a,b,c$ are in H.P. and $ab+bc+ca=15,$ then $ca=$

1. $5$
2. $7.5$
3. $3$
4. $4.5$

Q. The roots of the equation $x^{\frac{2}{3}}+x^{\frac{1}{3}}-2=0$, are

1. $1,4$
2. $1,-4$
3. $1,8$
4. $1,-8$

Q.

What Is Harmonic Progression Formula?

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. Prove by the Principle of Mathematical Induction: $n^2<2^n$for all natural numbers $n\ge 5$.

Q. Match List-I with List-II.

	List-I		List-II
P.	If $t_1,t_2,t_3,t_4,t_5$ are first five terms of an A.P. $\frac{4(t_1-t_2-t_4)+6t_3+t_5}{t_1}$ then is equal to $(t_1\ne 0)$	(1)	2
Q.	$\sum _{K=1}^{\infty }\sum _{n=1}^{\infty }\frac{K}{2^{n+K}}$is less than or equal to	(2)	3
R.	If $T_n$ denotes the nth term of the H.P. from the beginning such that $\frac{T_2}{T_6}=9,$ then the value of $\frac{T_5}{T_{12}}$ is divisible by	(3)	4
S.	The value of $\frac{2}{1^3}+\frac{6}{1^3+2^3}+\frac{12}{1^3+2^3+3^3}$$+\frac{20}{1^3+2^3+3^3+4^3}+..$ upto infinity is less than or equal to	(4)	5
		(5)	6
		(6)	1

1. P(2); Q(1, 2, 3, 4, 5); R(2, 6); S(3, 4, 5)
2. P(1); Q(2, 6); R(3, 4); S(4, 5)
3. P(2); Q(3, 4, 5); R(1, 2, 3, 4, 5); S(2, 6)
4. P(1); Q(2); R(2); S(3)

Q. If $p^{th},q^{th},r^{th}$ terms of an A.P. are $a,b,c$ respectively, then the value of $a(q-r)+b(r-p)+c(p-q)$ is

1. $1$
2. $0$
3. $-1$
4. $2$

Q.

If $\frac{a+bx}{a-bx}$ = $\frac{b+cx}{b-cx}$ = $\frac{c+dx}{c-dx}$(x ≠ 0), then a, b, c, d are in:

1. H.P.

2. A.G.P

3. A.P.

4. G.P.

Q. If the sum of a certain number of terms of the $A.P.25,22,19,...$ is $116.$ Find the last term.

Q. If $x,y,z$ are different and $\Delta =\left|\begin{array}{ccc}x& x^2& 1+x^3\\ y& y^2& 1+y^3\\ z& z^2& 1+z^3\end{array}\right|=0,$ then show that $1+xyz=0.$

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. 

2. 

3. 

4. 

Q. If $\overrightarrow{a}=\alpha \hat{i}+\beta \hat{j}+3\hat{k},$$\overrightarrow{b}=-\beta \hat{i}-\alpha \hat{j}-\hat{k}$ and
$\overrightarrow{c}=\hat{i}-2\hat{j}-\hat{k}$
such that $\overrightarrow{a}\cdot \overrightarrow{b}=1$ and $\overrightarrow{b}\cdot \overrightarrow{c}=-3$, then $\frac{1}{3}((\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{c})$ is equal to

Q. If $\alpha ,\beta$ are roots of $375x^2-25x-2=0$ and $S_n={\alpha }^n+{\beta }^n$, then $li{m}_{n\to \infty }{\sum }_{r=1}^n{S}_r$ is

1. $\frac{7}{116}$
2. $\frac{1}{12}$
3. $\frac{29}{398}$
4. $\frac{1}{2}$

Q.

What are AP and HP?

Q.

If x, y, z are in H.P., then the value of expression log(x + z) + log(x - 2y + z) will be

1. log(x - z)

2. 2log(x - z)

3. 3log(x - z)

4. 4log(x - z)