Answer
$(-1)^n7(15)^{n-2}$
$7(15)^{n-2}$
Work Step by Step
We are given the geometric sequence:
$a_2=7$
$a_4=1575$
Determine the common ratio $r$:
$\dfrac{a_4}{a_2}=\dfrac{1575}{7}$
$\dfrac{a_1r^3}{a_1r}=225$
$r^2=225$
$r=\pm\sqrt{225}$
$r=\pm 15$
Case 1: $r=-15$
Determine $a_1$:
$a_2=a_1r$
$7=a_1\cdot (-15)$
$a_1=-\dfrac{7}{15}$
Determine $a_n$:
$a_n=a_1r^{n-1}=-\dfrac{7}{15}(-15)^{n-1}=(-1)^n 7(15)^{n-2}$
Case 2: $r=15$
Determine $a_1$:
$a_2=a_1r$
$7=a_1\cdot 15$
$a_1=\dfrac{7}{15}$
Determine $a_n$:
$a_n=a_1r^{n-1}=\dfrac{7}{15}(15)^{n-1}=7(15)^{n-2}$
