Hi,
Find the genral term of the G.P with third term 1 and the seventh term 8?
Thanks!!!
Geometric Series
Btw, what does "genral" mean?
The general term of any geometric sequence is given by
Tn = $\displaystyle ar^{n-1}$
where clearly, $\displaystyle a$ is the first term and $\displaystyle r$ is the common ratio of the geometric progression.
So we have here,
T3=1
T7=8
i.e. $\displaystyle ar^2=1$............(1)
and$\displaystyle ar^6=8$............(2)
Divide (2) by (1) to eliminate a
so we have, $\displaystyle {r^6}/{r^2}$= 8/1
so $\displaystyle r^4=8$
$\displaystyle i.e. r = 8^{1/4}$
$\displaystyle Now using the fact that ar^2=1$
$\displaystyle we have, a(8)^{1/2}=1$
$\displaystyle so a = 8^{-1/2}$
$\displaystyle Hence the general term of the series is given by$
$\displaystyle Tn= 8^{-1/2}(8^{1/4})^{n-1}$