R D Sharma Solutions for Chapter: Geometric Progressions, Exercise 1: EXERCISE 20.1

Find the $4^{\mathrm{th}}$ term from the end of the G.P. $\frac{1}{2},\frac{1}{6},\frac{1}{18},\frac{1}{54},\dots ,\frac{1}{4374}$

If the G.P.'s $\mathrm{5,10,20},...$ and $\mathrm{1280,640,320},...$ have their $n^{th}$ terms equal, find the value of $n$.

If $5^{\mathrm{th}}, 8^{\mathrm{th}}$ and ${11}^{\mathrm{th}}$ terms of a G.P. are $p,q$ and $s$ respectively, prove that $q^2=ps$.

The$4th$ term of a $G.P$. is square of its second term, and the first term is - $3$ . Find its $7^{\text{th }}$ term.

In a GP the $\mathrm{}{3}^{\mathrm{r}\mathrm{d}}$ term is $24$ and the $6^{\mathrm{t}\mathrm{h}}$ teram is $192$. Find the ${10}^{\mathrm{t}\mathrm{h}}$ term.

If $a,b,c,d$and $p$ are different real numbers such that:

$\left(a^2+b^2+c^2\right)p^2-2\left(ab+bc+cd\right)p+\left(b^2+c^2+d^2\right)\le 0,$ then show that $a,b,c$ and $d$ are in G.P.

If $\frac{a+bx}{a-x}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}, \mathrm{with} x\ne 0$, then show that $a,b,c$ and $d$ are in G.P.

If the $p^{\mathrm{t}\mathrm{h}}$ and $q^{\mathrm{t}\mathrm{h}}$ terms of a G.P. are $q$ and $p$ respectively, show that ${\left(p+q\right)}^{\mathrm{t}\mathrm{h}}$ term is ${\left(\frac{q^p}{p^q}\right)}^{\frac{1}{p-q}}$.