

Use summation formulas to evaluate the sum: sum from k equals 1 to 26 of left parenthesis 4 k squared minus 19 k plus 7 right parenthesis.

    sum from k equals 1 to 26 of left parenthesis 4 k squared minus 19 k plus 7 right parenthesis equals 31480

    sum from k equals 1 to 26 of left parenthesis 4 k squared minus 19 k plus 7 right parenthesis equals 18317

    sum from k equals 1 to 26 of left parenthesis 4 k squared minus 19 k plus 7 right parenthesis equals 31655

    sum from k equals 1 to 26 of left parenthesis 4 k squared minus 19 k plus 7 right parenthesis equals 18142


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.

Use summation formulas to evaluate the sum: sum from k equals 1 to 26 of left parenthesis 4 k squared minus 19 k plus 7 right parenthesis.

Explore how to evaluate the summation of the expression 4k^2 - 19k + 7 from k=1 to 26 by applying summation formulas. This step-by-step solution covers key mathematical concepts in series and quadratic expressions, ideal for high school students in Grades 10-12.

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