Answer
$a_7 = \frac{4}{625}$
Work Step by Step
Use the explicit formula to find a specific term in a geometric sequence:
$a_n = a_1 \bullet r^{n - 1}$
Let's find the common ratio:
$r = \frac{20}{100} = \frac{1}{5}$
The common ratio $r$ is $\frac{1}{5}$. We are given that the first term of the series, $a_1$, is $100$. Substitute these values into the explicit formula to find the $7th$ term:
$a_7 = 100 \bullet (\frac{1}{5})^{7 - 1}$
Simplify the exponent:
$a_7 = 100 \bullet (\frac{1}{5})^{6}$
Evaluate the exponential term first:
$a_7 = 100 \bullet \frac{1}{15,625}$
Multiply to simplify:
$a_7 = \frac{100}{15,625}$
Simplify by dividing by the greatest common factor of the numerator and denominator, $25$:
$a_7 = \frac{4}{625}$
