Properties of GP

Q.

If $1+(1-2^2\times 1)+(1-4^2\times 3)+(1-6^2\times 5)+......+(1-{20}^2\times 19)=\alpha -220\beta$, then an ordered pair $(\alpha ,\beta )$ is equal to

1. $(10,97)$

2. $(11,103)$

3. $(11,97)$

4. $(10,103)$

Q.

Let $\alpha$ and $\beta$ be the roots of $x^2–x–1=0$, with $\alpha >\beta$. For all positive integers $n$, define

$\begin{array}{l}{a}_n=\frac{{\alpha }^n–{\beta }^n}{\alpha –\beta },n\ge 1\end{array}$

$b_1=1$ and $b_n=a_{n-1}+a_{n+1},n\ge 2.$

Then which of the following options is/are correct?

1. $\begin{array}{l}{a}_1+a_2+a_3+\dots .+a_n=a_{n+2}–1\end{array},n\ge 1$

2. $\begin{array}{l}\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{{10}^n}=\frac{10}{89}\end{array}$

3. $\begin{array}{l}\sum _{n=1}^{\mathrm{\infty }}\frac{b_n}{{10}^n}=\frac{8}{89}\end{array}$

4. $\begin{array}{l}{b}_n={\alpha }^n+{\beta }^n\end{array},n\ge 1$

Q.

If $a+b+c=0$, then the value of $(a+b-c{)}^3+(b+c-a{)}^3+(c+a-b{)}^3$ is:

1. $8(a^3+b^3+c^3)$

2. $a^3+b^3+c^3$

3. $24abc$

4. $-24abc$

Q.

The sum of a few terms of any ratio series is $728$, if the common ratio is $3$ and the last term is $486$, then the first term of the series will be

1. $2$

2. $1$

3. $3$

4. $4$

Q.

The ratio of one Micron to one Nanometre is,

1. ${10}^{-1}$

2. ${10}^3$

3. ${10}^{-3}$

4. ${10}^{-6}$

Q.

If $(a-b):(a+b)=1:11$ , find the ratio $(5a+4b+15):(5a-4b+3)$ .

Q.

How many terms of the series 3, 9, 27 . . . . . . . will add up to 363?

__

Q.

There is a natural number$x$. Write down the expression for the product of $x$and its next natural number.

1. (x + 1)(x+2)

2. x² + x

3. x² -x

4. 2x² +1

Q.

If $a,b,c$ are in A.P., $a,x,b$ are in GP and $b,y,c$ are also in G.P, then $x^2,b^2,y^2$ are in AP.

1. True

2. False

Q.

The number which should be added to 2, 14 and 62 so that the resulting numbers are in G.P. is ___.

1. 4

2. 1

3. 2

4. 3

Q.

Find the${10}^{\mathrm{th}}$ term of the GP : $12,4,\frac{4}{3},\frac{4}{9}$.

Q.

How many terms of the series $1+3+3^2+3^3+\dots$ must be taken to make $3280$.

Q.

If 5, 15, 45, 135, 405, 1215.... are in GP, which series is also in GP?

1. 5, 15, 405

2. 5, 45, 405

3. 5, 135, 405

4. 5, 15, 135

Q.

If the sum of the $3^3+7^3+{11}^3+{15}^3...................20$ terms is $S_{20}$. Find the value of $\frac{S_{20}}{100}$ .

__

Q.

In the given G.P find the product of the fifth term from the begining and from the end.

$\frac{1}{16},\frac{1}{4},1,4,\mathrm{16.........16},384.$

1. 256

2. 4096

3. 4056

4. 1024

Q.

The ${11}^{th},{13}^{th}\mathrm{and}{15}^{th}$ terms of any G.P are in:

1. G.P

2. A.P

3. H.P

4. A.G.P

Q. The sum of three numbers in G.P. is $38$, and their product is $1728$; find them.

Q.

If a, b, c, and d are in G. P., then $(a+b{)}^2$ , $(b+c{)}^2$ , and $(c+d{)}^2$ are also in G. P.

1. True

2. False

Q.

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}=30800:1$, then r = 41

1. 31

2. 41

Q.

If a, b and c are in A.P. and also in G.P., show that : a = b = c .

Q. If m is the AM of two distinct real numbers l and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals

1. $4l^{2}mn$
2. $4l{m}^{2}n$
3. $lm{n}^2$
4. $l^2{m}^2{n}^2$

Q.

How do you write $0.888...$ ($8$ repeating) as a fraction?

Q.

If a, b and c are in A.P, a, x, b are in G.P. whereas b, y and c are also in G.P. Show that : $x^2,b^2,y^2$ are in A.P.

Q. Find the total area of $14$ square whose sides are $11cm,12cm,.....24cm$, respectively.

Q.

If a - 14, a - 2, a + 34 are in G.P, find the value of a.

___

Q. If $\alpha ,\beta$ are the roots of $x^2-3x+a=0$ and $\gamma ,\delta$ that of $x^2-12x+b=0$ and $\alpha ,\beta ,\gamma ,\delta$ form an increasing G.P, then

1. $a=12,b=3$
2. $a=4,b=16$
3. $a=3,b=12$
4. $a=2,b=32$

Q.

The value of x for which the numbers 2x+1, 5x+1, and 11x+1 are in G.P is ___ (x is a natural number).

Q.

If a, b, c are in G.P, then log a, log b and log c are in ____

1. a(b² + c²) = c(a² + b²)
2. a²(b + c) = c²(a + b)
3. none of these
4. a(b² + c²) = c(b² + c²)

Q. How many terms of the series $1+4+16+$ .......... make the sum $1,365$?

Q. $a^2,b^2,c^2$ are in A.P
Then prove that $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in A.P