Finding the Value of Statement P(n) for the Series

The given statement P(n) represents the sum of a series:

$$P(n) = 1 \cdot 6 + 2 \cdot 9 + 3 \cdot 12 + \dots + n(3n + 3)$$

We can observe the pattern of the terms. The k-th term of the series is given by:

$$a_k = k(3k + 3)$$

We can simplify the k-th term:

$$a_k = 3k^2 + 3k$$

The statement P(n) is the sum of the first n terms of this series. So, we need to calculate the sum $S_n = \sum_{k=1}^{n} a_k$:

$$S_n = \sum_{k=1}^{n} (3k^2 + 3k)$$

Using the properties of summation, we can split this into two separate sums:

$$S_n = \sum_{k=1}^{n} 3k^2 + \sum_{k=1}^{n} 3k$$

We can factor out the constants from the summations:

$$S_n = 3 \sum_{k=1}^{n} k^2 + 3 \sum_{k=1}^{n} k$$

Now, we use the standard formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers:

• Sum of first n natural numbers: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
• Sum of squares of first n natural numbers: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

Substitute these formulas into the expression for $S_n$:

$$S_n = 3 \left( \frac{n(n+1)(2n+1)}{6} \right) + 3 \left( \frac{n(n+1)}{2} \right)$$

Simplify the first term:

$$S_n = \frac{n(n+1)(2n+1)}{2} + \frac{3n(n+1)}{2}$$

Now, we have a common denominator. We can combine the terms:

$$S_n = \frac{n(n+1)(2n+1) + 3n(n+1)}{2}$$

Factor out the common term $n(n+1)$ from the numerator:

$$S_n = \frac{n(n+1)[(2n+1) + 3]}{2}$$

Simplify the expression inside the square brackets:

$$S_n = \frac{n(n+1)[2n + 4]}{2}$$

Factor out 2 from the term in the square brackets:

$$S_n = \frac{n(n+1) \cdot 2(n+2)}{2}$$

Cancel out the 2 in the numerator and denominator:

$$S_n = n(n+1)(n+2)$$

Thus, the value of statement P(n) is $n(n+1)(n+2)$.

Let's verify with a small value, say n=1:

• Series sum for n=1 is the first term: $1 \cdot 6 = 6$.
• Our formula gives: $1(1+1)(1+2) = 1 \cdot 2 \cdot 3 = 6$. The formula holds for n=1.

Let's verify with n=2:

• Series sum for n=2 is the sum of the first two terms: $1 \cdot 6 + 2 \cdot 9 = 6 + 18 = 24$.
• Our formula gives: $2(2+1)(2+2) = 2 \cdot 3 \cdot 4 = 24$. The formula holds for n=2.

The derived formula matches the required value of statement P(n).