To evaluate the summation:

$\sum_{i=1}^{27} \left(i^3 - 3t^2 + 5i\right),$

let's break it into parts. The summation can be separated into individual sums:

$\sum_{i=1}^{27} \left(i^3 - 3t^2 + 5i\right) = \sum_{i=1}^{27} i^3 - \sum_{i=1}^{27} 3t^2 + \sum_{i=1}^{27} 5i.$

Step 1: Evaluate $\sum_{i=1}^{27} i^3$

The formula for the sum of cubes is:

$\sum_{i=1}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2.$

For $n = 27$:

$\sum_{i=1}^{27} i^3 = \left(\frac{27(27+1)}{2}\right)^2 = \left(\frac{27 \cdot 28}{2}\right)^2 = (378)^2 = 142884.$

Step 2: Evaluate $\sum_{i=1}^{27} 3t^2$

Since $3t^2$ is independent of $i$, it can be factored out of the summation. There are 27 terms:

$\sum_{i=1}^{27} 3t^2 = 27 \cdot 3t^2 = 81t^2.$

Step 3: Evaluate $\sum_{i=1}^{27} 5i$

The formula for the sum of the first $n$ integers is:

$\sum_{i=1}^n i = \frac{n(n+1)}{2}.$

For $n = 27$:

$\sum_{i=1}^{27} 5i = 5 \cdot \frac{27(27+1)}{2} = 5 \cdot \frac{27 \cdot 28}{2} = 5 \cdot 378 = 1890.$

Combine the Results

Now, substitute the individual sums back into the original expression:

$\sum_{i=1}^{27} \left(i^3 - 3t^2 + 5i\right) = \sum_{i=1}^{27} i^3 - \sum_{i=1}^{27} 3t^2 + \sum_{i=1}^{27} 5i.$

$= 142884 - 81t^2 + 1890.$

Final Answer:

$\boxed{144774 - 81t^2}.$

Would you like further steps broken down or questions answered?

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Related Questions:

1. How do you derive the formula for the sum of cubes?
2. What is the general approach to split and simplify summations?
3. How does the summation change if $t$ is a function of $i$?
4. Can this sum be simplified for large $n$ using approximations?
5. How does the result change if the summation limits are altered?

Tip: Always check if terms in a summation can be separated into simpler independent parts for easier calculation.