Common Ratio

Q.

Find the ${12}^{th}$ term of a G.P. whose $8^{th}$ term is $192$ and the common ratio is $2$.

Q.

Find the 8th term from the end of the GP 3, 6, 12, 24, ..., 12288.

Q.

Insert two numbers between 3 and 81 so that the resulting sequences is G.P.

Q. Show that $9^{n+1}-8n-9$ is divisible by 64 whenever n is a positive integer.

Q.

Three numbers a, b, c are in G.P. Their sum is 26 and their product is 216. Find the common ratio [integer value].

Q. If the sum of three terms of G.P. is 19 and product is 216, then the common ratio of the series is
[Roorkee 1972]

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

Q.

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that $P^2{R}^n=S^n$

Q.

The sides of a triangle are in AP and its area is 3/5 x (area of an equilateral triangle of the same perimeter) Then, the ratio of the sides is

1. 3:5:7

2. 1:3:5

3. None of these

4. 1:2:3

Q.

The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.

Q. If the normal at the end of latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccen-tricity e passes through one end of the minor axis then $e^4+e^2=$

1. 
2. 
3. 
4. 

Q.

If the first term of a G.P. be 5 and common ratio be -5, then which term is 3125

1. 

2. 

3. 

4. 

Q. The sum of first three terms of a G.P. is $16$ and the sum of next three terms is $128$. Determine the first term, the common ratio and the sum to $n$ terms of the G.P.

Q. the ratio of the sum of n terms of two APs is (3n+1):(4n+3). find the ratio of their mth terms

Q. The sum of three numbers in G.P. is $56$. If we subtract $1,7,21$ from these numbers in that order, we obtain an arithmetic progression . Find the numbers

Q. If $a,b,c$ are in G.P. prove that $(a^n+b^n),(b^n+c^n),(c^n+d^n)$ are in G.P.

Q. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Q.

If the 2nd, 5th and 9th terms of a non-constant AP are in GP, then the common ratio of this GP is

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

2. $\frac{8}{5}$

3. $1$

4. $\frac{7}{4}$

Q.

Show that the products of the corresponding terms of the sequences $a,ar,a{r}^2,.......a{r}^{n-1}andA,AR,A{R}^2,....A{R}^{n-1}$ form a G.P and find the common ratio.

Q. The sum of first 10 terms of the G.P 3, 6, 12, 24, ... is .

1. 3500
2. 3069
3. 2059
4. 4079

Q.

$\frac{1}{2}$, $\frac{1}{4}$, - - - - form a H.P The ratio of 5th term to 7th term is

1. 

2. 

3. 

4. 

Q. The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is $1$. Find the common ratio and the terms.

Q. If p^th, q^th and r^th terms of a G.P. are x, y, z respectively, then write the value of x^{q − r} y^{r − p} z^{p − q}.

Q.

Three positive numbers form an increasing G.P. If the middle term in this G.P is doubled, then new numbers are in A.P. Then, the common ratio of the G.P. is

1. $2+\sqrt{3}$

2. $2-\sqrt{3}$

3. $\sqrt{2}+\sqrt{3}$

4. $3+\sqrt{2}$

Q.

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from $(n+1{)}^{th}to(2n{)}^{th}$term is $\frac{1}{r^n}$

Q.

The sum of three numbers in GP is 56. If we subtract 1, 7, 21 from these numbers in that order, we get an AP. Find the numbers.

Q. The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Q.

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.

Q. The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is

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

Q. If the sum of three terms of G.P. is 19 and product is 216, then the common ratio of the series is
[Roorkee 1972]

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

Q. Three positive numbers form an increasing G.P. If the middle term in this G.P is doubled, then new numbers are in A.P. Then, the common ratio of the G.P. is

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