To solve the sum [tex]\(\sum_{n=1}^4 \left(3n^3 + 1\right)\)[/tex], let's break it down step-by-step.
First, we need to understand the expression inside the summation:
[tex]\[ f(n) = 3n^3 + 1 \][/tex]
Now, we will evaluate this expression for each integer value of [tex]\( n \)[/tex] from 1 to 4, and then sum the results.
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[ f(1) = 3(1)^3 + 1 = 3(1) + 1 = 3 + 1 = 4 \][/tex]
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[ f(2) = 3(2)^3 + 1 = 3(8) + 1 = 24 + 1 = 25 \][/tex]
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[ f(3) = 3(3)^3 + 1 = 3(27) + 1 = 81 + 1 = 82 \][/tex]
4. For [tex]\( n = 4 \)[/tex]:
[tex]\[ f(4) = 3(4)^3 + 1 = 3(64) + 1 = 192 + 1 = 193 \][/tex]
Next, we add up all these values:
[tex]\[
f(1) + f(2) + f(3) + f(4) = 4 + 25 + 82 + 193
\][/tex]
Finally, summing these numbers:
[tex]\[
4 + 25 = 29
\][/tex]
[tex]\[
29 + 82 = 111
\][/tex]
[tex]\[
111 + 193 = 304
\][/tex]
Thus, the result of the sum [tex]\(\sum_{n=1}^4 \left(3n^3 + 1\right)\)[/tex] is:
[tex]\[
\boxed{304}
\][/tex]
