1.2.3 + 2.3.4 + 3.4.5 +…n terms is `(n (n + 1) (n + 2) (n + 3))/(P)` where P =

Options

A. 4
B. 6
C. 8
D. 12

Answer

To solve the problem, we need to find the sum of the series \(1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + 3 \cdot 4 \cdot 5 + \ldots\) up to \(n\) terms and express it in the form \(\frac{n(n + 1)(n + 2)(n + 3)}{P}\), where \(P\) is to be determined.

### Step 1: Identify the general term
The general term of the series can be expressed as:
\[ T_n = n(n + 1)(n + 2) \]
This represents the \(n\)-th term of the series.

### Step 2: Write the sum of the series
The sum of the first \(n\) terms can be written as:
\[ S_n = \sum_{k=1}^{n} k(k + 1)(k + 2) \]

### Step 3: Simplify the general term
We can rewrite the general term:
\[ T_k = k(k + 1)(k + 2) = k^3 + 3k^2 + 2k \]
Thus, the sum becomes:
\[ S_n = \sum_{k=1}^{n} (k^3 + 3k^2 + 2k) \]

### Step 4: Use summation formulas
We can use the formulas for the sums of powers:
- \(\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}\)
- \(\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}\)
- \(\sum_{k=1}^{n} k^3 = \left(\frac{n(n + 1)}{2}\right)^2\)

Using these, we can compute \(S_n\):
\[ S_n = \sum_{k=1}^{n} k^3 + 3\sum_{k=1}^{n} k^2 + 2\sum_{k=1}^{n} k \]

### Step 5: Substitute the summation results
Substituting the summation results into the equation:
\[ S_n = \left(\frac{n(n + 1)}{2}\right)^2 + 3\left(\frac{n(n + 1)(2n + 1)}{6}\right) + 2\left(\frac{n(n + 1)}{2}\right) \]

### Step 6: Simplify the expression
After substituting and simplifying, we can find that:
\[ S_n = \frac{n(n + 1)(n + 2)(n + 3)}{4} \]

### Step 7: Identify \(P\)
From the expression for \(S_n\), we can see that it is in the form:
\[ S_n = \frac{n(n + 1)(n + 2)(n + 3)}{P} \]
Comparing both sides, we find that \(P = 4\).

### Final Answer
Thus, the value of \(P\) is:
\[ \boxed{4} \]