Selecting Consecutive Terms in GP

Q.

Find three numbers in G.P. whose sum is 38 and their product is 1728.

Q.

The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

Q.

The sum of first three terms of a G.P. is $\frac{13}{12}$ and their product is - 1. Find the G.P.

Q.

The sum of first two terms of a G.P is $1$ and every term of this series is twice of its previous term, then the first term will be

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

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

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

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

Q.

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

Q.

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}$. Find them.

Q. In a G.P. the sum of three numbers is 14, if 1 is added to first two numbers and subtracted from third number, the series becomes A.P., then the greatest number is

1. 8
2. 4
3. 24
4. 16

Q.

The value of p and q$p\ne 0,q\ne 0$ for which p, q are the roots of the equation: $x^2+px+q=0$ are

1. p = 1, q = 2

2. p = -1, q =2

3. p = 1, q = -2

4. p = -1, q = -2

Q.

Find three numbers in G.P. whose sum is 65 and whose product is 3375.

Q.

If a and b are the roots of $x^2-3x+p=0$ and c, d are the roots $x^2-12x+q=0$, Where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15.

Q.

If (a - b), (b - c), (c - a) are in G.P. then prove that $(a+b+c{)}^2=3(ab+bc+ca)$

Q.

The sum of three numbers in G.P. is 14, If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.

Q.

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < -1 or x > 3.

Q.

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

Q. If $f(x+y)=f\left(x\right)f\left(y\right)$ and $\sum _{x=1}^{\infty }f\left(x\right)=2,x,y\in N,$ where $N$ is the set of all natural numbers, then the value of $\frac{f\left(4\right)}{f\left(2\right)}$ is:

1. $\frac{2}{3}$
2. $\frac{1}{9}$
3. $\frac{1}{3}$
4. $\frac{4}{9}$

Q. Three numbers form a G.P. If the $3^{rd}$ term is decreased by 64, then the three numbers thus obtained will constitute an A.P. If the second term of this A.P. is decreased by 8, a G.P. will be formed again, then the numbers will be

1. 4, 20, 36
2. 4, 12, 36
3. 4, 20, 100
4. None of the above

Q.

If $a,4,b$are in $AP$ and $a,2,b$are in $GP$ then $\frac{1}{a},1,\frac{1}{b}$ are in

1. $AP$

2. $GP$

3. $HP$

4. none of these

Q. Let $a,b,c$ be three distinct real numbers in geometric progression. If $x$ is real and $a+b+c=xb,$ then $x$ can be

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

Q.

If a, b, c are in G.P. then prove that : $\frac{a^2+ab+b^2}{bc+ca+ab}=\frac{b+a}{c+b}$

Q. If the ${10}^{th}$ term of a GP is 9 and $4^{th}$ term is 4, then its $7^{th}$ term is

1. 
2. 
3. 
4. 

Q. If three distinct real numbers a, b, c are in G.P and $a+b+c=ax$, then

1. 
2. 
3. x ∊ (-1, ∞)
4. none of these

Q. Given the geometric progression $3,6,12,24,....$ The term $12288$ would occur as the

1. ${11}^{th}term$
2. ${12}^{th}term$
3. ${13}^{th}term$
4. ${14}^{th}term$

Q.

The product of three numbers in $G.P.$ is $216.$ If $2,8,6$ be added to them$,$ the results are in $A.P.$ Find the numbers$.$

Q. Sum of $\frac{1}{3},\frac{1}{15},\frac{1}{35},\frac{1}{63},\dots \dots 2n$ terms is

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

Q.

Find the sum of the GP $(1+x{)}^{21}+(1+x{)}^{22}+......(1+x{)}^{30}$

1. 

2. 

3. 

4. 

Q. If the geometric sequences $162,54,18,....$ and $\frac{2}{81},\frac{2}{27},\frac{2}{9},....$ have their $n^{th}$ term equal, find the value of $n$

Q. If $5p^2-7p-3=0$ and $5q^2-7q-3=0$, $p\ne q$, then the equation whose roots are $5p-4q$ and $5q-4p$ is :

1. $5x^2+x-439=0$
2. $5x^2+7x-439=0$
3. $5x^2-7x-439=0$
4. $5x^2+7x+439=0$

Q. Let $a,b,c$ be three distinct real numbers in geometric progression. If $x$ is real and $a+b+c=xb,$ then $x$ can be

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

Q. If the product of three consecutive term in $G.P$ is $216$ and sum of their products in pairs is $156$, find them.

Q. If x, $G_1,G_2,$ y be the consecutive terms of a G.P., then the value of $G_1{G}_2$ will be

1. $\frac{y}{x}$
2. $\frac{x}{y}$
3. xy
4. $\sqrt{xy}$