Find the $20^{\text {th }}$ and $n^{\text {th }}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$
Find the $20^{\text {th }}$ and $n^{\text {th }}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$
The given $G.P.$ is $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots .$
Here, $a=$ First term $=\frac{5}{2}$
$r=$ Common ratio $=\frac{5 / 4}{5 / 2}=\frac{1}{2}$
$a_{20}=a r^{20-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{19}=\frac{5}{(2)(2)^{19}}=\frac{5}{(2)^{20}}$
$a_{n}=a r^{n-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{n-1}=\frac{5}{(2)(2)^{n-1}}=\frac{5}{(2)^{n}}$
Similar Questions
The two geometric means between the number $1$ and $64$ are
The two geometric means between the number $1$ and $64$ are
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
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
If $a,\,b,\,c$ are in $A.P.$ and ${a^2},\,{b^2},{c^2}$ are in $H.P.$, then
If $a,\,b,\,c$ are in $A.P.$ and ${a^2},\,{b^2},{c^2}$ are in $H.P.$, then
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 $(2 n)^{ th }$ term is $\frac{1}{r^{n}}$
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 $(2 n)^{ th }$ term is $\frac{1}{r^{n}}$
Let $n \geq 3$ and let $C_1, C_2, \ldots, C_n$, be circles with radii $r_1, r_2, \ldots, r_n$, respectively. Assume that $C_i$ and $C_{i+1}$ touch externally for $1 \leq i \leq n-1$. It is also given that the $X$-axis and the line $y=2 \sqrt{2} x+10$ are tangential to each of the circles. Then, $r_1, r_2, \ldots, r_n$ are in
Let $n \geq 3$ and let $C_1, C_2, \ldots, C_n$, be circles with radii $r_1, r_2, \ldots, r_n$, respectively. Assume that $C_i$ and $C_{i+1}$ touch externally for $1 \leq i \leq n-1$. It is also given that the $X$-axis and the line $y=2 \sqrt{2} x+10$ are tangential to each of the circles. Then, $r_1, r_2, \ldots, r_n$ are in
- [KVPY 2014]