M L Aggarwal Solutions for Chapter: Sequences and Series, Exercise 7: EXERCISE 10.7

If $G_1$ and $G_2$ are two $G.M.\text{'}s$ between two positive numbers $a$ and $b$, then show that $G_1{G}_2=ab$.

If $A_1,A_2$ are two $A.M\text{'}s$ and $G_1,G_2$ are two $G.M\text{'}s$ between two positive numbers $a$ and $b$, then find the value of $\frac{A_1+A_2}{G_1{G}_2}$.

Insert $5$ geometric means between $576$ and $9$. Show that their product is equal to the fifth power of the single $G.M.$ between the given numbers.

The arithmetic mean between two numbers is $34$ and $G.M.$ is $16$. Find the numbers.

The $A.M.$ of two numbers $a$ and $b$ exceeds their G.M. by $2$. The ratio of two numbers is $4$. Find the numbers.

Find two positive numbers whose difference is $12$ and whose $A.M.$ exceeds $G.M.$ by $2$.

The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller.

If $G$ is the geometric mean between $a$ and $b$, show that $\frac{1}{G+a}+\frac{1}{G+b}=\frac{1}{G}$.