To evaluate this summation using summation formulas, we can break it down and use known summation formulas for $k$, $k^2$, and constants. The expression is:

$\sum_{k=1}^{26} (4k^2 - 19k + 7)$

This can be rewritten as:

$4 \sum_{k=1}^{26} k^2 - 19 \sum_{k=1}^{26} k + \sum_{k=1}^{26} 7$

We'll evaluate each of these components separately.

Step 1: Sum of $k^2$ Terms

The formula for the sum of squares, $\sum_{k=1}^n k^2$, is: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ For $n = 26$: $\sum_{k=1}^{26} k^2 = \frac{26 \times 27 \times 53}{6} = 6201$ Then, $4 \sum_{k=1}^{26} k^2 = 4 \times 6201 = 24804$

Step 2: Sum of $k$ Terms

The formula for the sum of integers, $\sum_{k=1}^n k$, is: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ For $n = 26$: $\sum_{k=1}^{26} k = \frac{26 \times 27}{2} = 351$ Then, $-19 \sum_{k=1}^{26} k = -19 \times 351 = -6669$

Step 3: Sum of Constant Terms

Since 7 is a constant, the sum of 26 terms of 7 is: $\sum_{k=1}^{26} 7 = 7 \times 26 = 182$

Step 4: Combine Results

Now, we add the three parts together: $4 \sum_{k=1}^{26} k^2 - 19 \sum_{k=1}^{26} k + \sum_{k=1}^{26} 7 = 24804 - 6669 + 182 = 18317$

Final Answer

The correct answer is: $\sum_{k=1}^{26} (4k^2 - 19k + 7) = 18317$

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Do you want more details on any of the steps or have any questions?

Related Questions

1. How can the sum of squares formula be derived?
2. What is the summation formula for $k^3$ and when is it useful?
3. Can summation formulas be applied to non-polynomial functions?
4. How can we generalize this approach for any quadratic polynomial?
5. What is the difference between arithmetic and geometric summation?

Tip:

When summing expressions with $k^2$ or $k^3$ terms, always break down the expression into simpler parts and use known formulas to simplify computation.