Geometric Progression

Q.

Find the sum: $1+2+3+4+5+......+100$.

Q.

Two numbers are in the ratio $5:6$. If $8$ is subtracted from each of the numbers, the ratio becomes $4:5$ find the numbers.

Q. A sheet of area $40m^2$ is used to make an open tank with square base. The side of the base such that the volume of this tank is maximum, is

1. $\sqrt{\frac{40}{3}}m$
2. $\sqrt{\frac{20}{3}}m$
3. $\sqrt{\frac{80}{3}}m$
4. $\sqrt{\frac{10}{3}}m$

Q. Which term of the sequence $2,1,\frac{1}{2},\frac{1}{4},\dots$ is $\frac{1}{128}$

1. $9$
2. $8$
3. $10$
4. $7$

Q.

If $a^2+b^2+c^2=1$, then ab + bc + ca lies in :

1. 

2. 

3. 

4. 

Q. Let $a_1,a_2,a_3\cdots ,a_{21}$ be an $AP$ such that $\sum _{n=1}^{20}\frac{1}{a_n{a}_{n+1}}=\frac{4}{9}$ If the sum of this $AP$ is $189$, then $a_6{a}_{16}$ is equal to:

1. $36$
2. $57$
3. $72$
4. $48$

Q.

Solve $23x-29y=98$And $29x-23y=110$

Q. If $m$ and $n$ are whole numbers such that $m^n=121,$ the value of $(m-1{)}^{n+1}$ is

1. $1$
2. $10$
3. $100$
4. $1000$

Q.

If $a_1,a_2,......a_n$are in arithmetic progression, where $a_i>0$ for all $i$. Then, $\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+.............+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}$

1. $n^2(n+1)/2$

2. $\frac{(n-1)}{\sqrt{a_1}+\sqrt{a_n}}$

3. $n(n-1)/2$

4. None of these

Q.

The $6$th term of the G.P is $32$ and its $8$th term is $128$, then the common ratio of the G.P is:

1. $-1$

2. $2$

3. $4$

4. $-4$

Q.

Let $P$ and $Q$ be $3\times 3$matrices, $P\ne Q$.If $P^3=Q^3$and $P^{2}Q=Q^{2}P,$ then the determinant of $(P^2+Q^2)$ is equal to

1. $-2$

2. $1$

3. $0$

4. $-1$

Q.

The value of $\sum _{0\le i<}^{}\sum _{j\le n}^{}({}^n{C}_i+{}^n{C}_j)is:$

1. 

2. 

3. 

4. 

Q.

If a, b, c, d are in G.p., prove that :

(i) $(a^2+b^2),(b^2+c^2),(c^2+d{)}^2$ are in G.P.

(ii) $(a^2-b^2),(b^2-c^2),(c^2-d{)}^2$ are in G.P.

(iii) $\frac{1}{a^2+b^2},\frac{1}{b^2+c^2},\frac{1}{c^2+d^2}$ are in G.P.

(iv) $(a^2+b^2+c^2),(ab+bc+cd),(b^2+c^2+d^2)$

Q.

If $\alpha ,\beta$ are the real and distinct roots of $x^2$ + px + q = 0 and ${\alpha }^4,{\beta }^4$ are the roots of $x^2-rx+5=0$, then the equation $x^2-4qx+2a^2-r=0$ has always

1. two real roots

2. two negative roots

3. two positive roots

4. one positive root and one negative root

Q.

If $a,b,c$ are in $G.P$., $a-b,c-a,b-c$are in $H.P$, then$a+4b+c$ is equal to

Q. If the fourth, seventh and the last term of a G.P. are $10,80$ and $2560$ respectively, then

1. first term is $\frac{5}{2}$
2. common ratio is $2$
3. number of terms is $12$
4. ${10}^{th}$ term is $640$

Q. In an $A.P.,$ if $p^{th}$ term is $\frac{1}{q}$ and $q^{th}$ term is $\frac{1}{p}$, prove that the sum of first $pq$ terms is $\frac{1}{2}(pq+1)$, where $p\ne q$.

Q.

If $A$ and $B$ are two sets, then $A\times B=B\times A$ if

1. $A\subseteq B$

2. $B\subseteq A$

3. $A=B$

4. None of these

Q. The $n^{\text{th}}$ term of the series $3,\sqrt{3},1,\dots$ is $\frac{1}{243},$ then $n=$

Q.

The sum of three consecutive odd integers is $99$. What are the three numbers?

Q.

What Are The $4$ Types Of Sequence?

Q.

If twice the ${11}^{\mathrm{th}}$ term of an A.P is equal to $7$ times of its $21\mathrm{st}$ term, then its $25\mathrm{th}$ term is equal to

1. $24$

2. $120$

3. $0$

4. None of these

Q. Let $a,b$ and $c$ be the $7^{\text{th}},{11}^{\text{th}}$ and ${13}^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to :

1. $4$
2. $2$
3. $\frac{7}{13}$
4. $\frac{1}{2}$

Q.

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. then $P^2{R}^3:S^3$ is equal to

1. 1 : 1

2. $($Common ratio${)}^n:1$

3. $($ First term ${)}^2$ $($ Common ratio ${)}^2$

4. None of these

Q.

If the sum of three terms of G.P is $19$ and product is $216$, then the common ratio of the series is

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

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

3. $2$

4. $3$

Q.

The sum of three consecutive terms in a geometric progression is $14$.

If $1$ is added to the first and the second terms and $1$ is subtracted from the third, the resulting new terms are in arithmetic progression.

Then the lowest of original term is

1. $1$

2. $2$

3. $4$

4. $8$

Q. Three non-zero numbers $a,b$ and $c$ are in A.P.. If we increase $a$ by $1$ or increase $c$ by $2,$ then the numbers are in G.P.. The value of $b$ is

Q.

Find the sum of the first $22$ terms of an AP in which $d=7$ and the $22nd$ term is $149$.

Q.

If the sum of first two terms of an infinite GP is 1 and every term is twice the sum of all the successive terms, then its first term is

Q.

Find the 4th term form the end of the G.P. $\frac{2}{27},\frac{2}{9},\frac{2}{3},.....,162$