Answer
$1$
Work Step by Step
Step 1. Identify the first term of the sequence which is the area of the first blue square: $a_1=(\frac{1}{3})^2$
Step 2. Calculate the second term which is the sum of $8$ small squares with $\frac{1}{3^2}$ of side length: $a_2=8\times(\frac{1}{3^2})^2$
Step 3. Calculate the third term which is the sum of $8^2$ small squares with $\frac{1}{3^3}$ of side length: $a_3=8^2\times(\frac{1}{3^3})^2$
Step 3. Write the $n$th term which is the sum of $8^{n-1}$ small squares with $\frac{1}{3^n}$ of side length: $a_n=8^{n-1}\times(\frac{1}{3^n})^2$
Step 4. It can be identified that $a_n$ forms a geometric sequence with $r=\frac{8}{9}$
Step 5. Calculate the total area of all the blue squares: $S_n=\frac{a_1}{1-r}=\frac{1/9}{1-8/9}=1$
