Answer :

To find the sum of the series

[tex]\[ \sum_{k=1}^4(3k)^2, \][/tex]

let's break it down step by step.

1. We start by substituting [tex]\( k \)[/tex] with 1, 2, 3, and 4 into the expression [tex]\( (3k)^2 \)[/tex].

2. For [tex]\( k = 1 \)[/tex]:
[tex]\[ (3 \cdot 1)^2 = 3^2 = 9. \][/tex]

3. For [tex]\( k = 2 \)[/tex]:
[tex]\[ (3 \cdot 2)^2 = 6^2 = 36. \][/tex]

4. For [tex]\( k = 3 \)[/tex]:
[tex]\[ (3 \cdot 3)^2 = 9^2 = 81. \][/tex]

5. For [tex]\( k = 4 \)[/tex]:
[tex]\[ (3 \cdot 4)^2 = 12^2 = 144. \][/tex]

6. Now, we need to add up all these values.
[tex]\[ 9 + 36 + 81 + 144. \][/tex]

7. Let's sum these values step by step:
[tex]\[ 9 + 36 = 45, \][/tex]

[tex]\[ 45 + 81 = 126, \][/tex]

[tex]\[ 126 + 144 = 270. \][/tex]

Therefore, the sum of the series is

[tex]\[ \boxed{270}. \][/tex]