
Geometric Sequences..
I don't understand how the book got the answer of 5/9
The question is:
Given the two terms of the geometric sequence, find the individual term.
t4=5 t8=405 find t2
i used the formula tn = ar^n1 for this
t8/t4 = 405/5 = ar^7 / ar^3
=81 = ar^4
r=3
5=a(3)^4
5=81a
a = 5/81
t2=(5/81)(3)^1
=5/27

You should know that $\displaystyle \displaystyle t_n = a\,r^{n  1}$.
Therefore $\displaystyle \displaystyle t_4 = a\,r^3$ and $\displaystyle \displaystyle t_8 = a\,r^7$.
From this we have
$\displaystyle \displaystyle a\,r^7 = 405$
$\displaystyle \displaystyle a\,r^3 = 5$.
Dividing the first equation by the second gives
$\displaystyle \displaystyle r^4 = 81$
$\displaystyle \displaystyle r = \pm 3$.
Substituting back into one of the equations:
$\displaystyle \displaystyle 405 = a\,(\pm 3)^7$
$\displaystyle \displaystyle 405 = \pm 2187 a$
$\displaystyle \displaystyle a = \pm\frac{5}{27}$.
So there are two possible sequences, the first with $\displaystyle \displaystyle a = \frac{5}{27}$ and $\displaystyle \displaystyle r = 3$, the second with $\displaystyle \displaystyle a = \frac{5}{27}$ and $\displaystyle \displaystyle r = 3$.
Either way, you should find that
$\displaystyle \displaystyle t_2 = a\,r = \frac{5}{27}\cdot 3 = \frac{5}{9}$ or $\displaystyle \displaystyle t_2 = \frac{5}{27}\left(3\right) = \frac{5}{9}$.