For any integer `n ge 1`, the sum `sum_(k=1)^(n) k (k + 2)` is equal to

Options

A. `(n (n + 1) (2n + 1))/(6)`
B. `(n(n + 1) (n + 2))/(6)`
C. `(n (n + 1)(2n + 7))/(6)`
D. none of these

Answer

To solve the problem of finding the sum \( S = \sum_{k=1}^{n} k(k + 2) \), we can break it down step by step.

### Step 1: Expand the Summation
We start by expanding the term inside the summation:
\[ S = \sum_{k=1}^{n} k(k + 2) = \sum_{k=1}^{n} (k^2 + 2k) \]

### Step 2: Split the Summation
Next, we can split the summation into two separate summations:
\[ S = \sum_{k=1}^{n} k^2 + \sum_{k=1}^{n} 2k \]

### Step 3: Use Summation Formulas
Now we will use the formulas for the sums of the first \( n \) natural numbers and the sum of the squares of the first \( n \) natural numbers:
- The formula for the sum of the first \( n \) natural numbers is:
\[ \sum_{k=1}^{n} k = \frac{n(n + 1)}{2} \]
- The formula for the sum of the squares of the first \( n \) natural numbers is:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 4: Substitute the Formulas
Substituting these formulas into our expression for \( S \):
\[ S = \frac{n(n + 1)(2n + 1)}{6} + 2 \cdot \frac{n(n + 1)}{2} \]

### Step 5: Simplify the Expression
Now we simplify the second term:
\[ 2 \cdot \frac{n(n + 1)}{2} = n(n + 1) \]
Thus, we can rewrite \( S \) as:
\[ S = \frac{n(n + 1)(2n + 1)}{6} + n(n + 1) \]

### Step 6: Combine the Terms
To combine the terms, we need a common denominator:
\[ S = \frac{n(n + 1)(2n + 1)}{6} + \frac{6n(n + 1)}{6} \]
This gives us:
\[ S = \frac{n(n + 1)(2n + 1 + 6)}{6} = \frac{n(n + 1)(2n + 7)}{6} \]

### Final Result
Thus, the final result for the sum \( S \) is:
\[ S = \frac{n(n + 1)(2n + 7)}{6} \]