Questions: (prodk=m^n(k+3)) cdot(prodk=m^n(k-2))

Transcript text: $\left(\prod_{k=m}^{n}(k+3)\right) \cdot\left(\prod_{k=m}^{n}(k-2)\right)$

Solution

Solution Steps

To solve the given expression, we need to evaluate the product of two sequences. The first sequence is the product of terms from $k = m$ to $k = n$ of $(k+3)$, and the second sequence is the product of terms from $k = m$ to $k = n$ of $(k-2)$. We can compute each product separately and then multiply the results.

Step 1: Evaluate the Products

We need to evaluate the two products separately:

The first product is given by: \[ P_1 = \prod_{k=1}^{5} (k + 3) \] This expands to: \[ P_1 = (1 + 3)(2 + 3)(3 + 3)(4 + 3)(5 + 3) = 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \]
The second product is: \[ P_2 = \prod_{k=1}^{5} (k - 2) \] This expands to: \[ P_2 = (1 - 2)(2 - 2)(3 - 2)(4 - 2)(5 - 2) = (-1)(0)(1)(2)(3) \]

Step 2: Calculate Each Product

Calculating $P_1$: \[ P_1 = 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 = 6720 \]

Calculating $P_2$: \[ P_2 = (-1) \cdot 0 \cdot 1 \cdot 2 \cdot 3 = 0 \]

Step 3: Combine the Results

Now, we combine the results of the two products: \[ \text{Result} = P_1 \cdot P_2 = 6720 \cdot 0 = 0 \]