Answer
$\dfrac{1093}{2187}$
Work Step by Step
We have to determine the sum:
$\sum_{k=1}^7 \left(\dfrac{1}{3}\right)^k$
Consider the geometric sequence:
$a_1=\left(\dfrac{1}{3}\right)^1=\dfrac{1}{3}$
$r=\dfrac{1}{3}$
Compute the sum of the first 7 terms:
$S_n=a_1\dfrac{1-r^n}{1-r}$
$n=7$
$\sum_{k=1}^7 \left(\dfrac{1}{3}\right)^k=S_7=\dfrac{1}{3}\cdot\dfrac{1-\left(\dfrac{1}{3}\right)^7}{1-\dfrac{1}{3}}$
$=\dfrac{1}{3}\cdot \left(1-\dfrac{1}{2187}\right)\cdot\dfrac{3}{2}=\dfrac{2186}{2(2187)}=\dfrac{1093}{2187}$