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]
