Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve for \(n\). $$4 \cdot_{n+1} P_{2}=_{n+2} P_{3}$$

Step-by-Step Solution

Answer

The solution to the equation is \(n = -\frac{2}{3}\)

Step 1: Understand and Rewrite the Given Equation

The given equation is \(4 \cdot_{n+1} P_{2}=_{n+2} P_{3}\). Rewrite this using the permutations formula: \(4 \cdot \frac{(n+1)!}{(n+1-2)!} = \frac{(n+2)!}{(n+2-3)!}\)

Step 2: Simplify the Equation

By simplifying, you get \(4 \cdot \frac{(n+1)!}{(n-1)!} = \frac{(n+2)!}{(n-1)!}\). The \((n-1)!\) on both sides can cancel out, leaving us with \(4 \cdot (n+1) = n+2\).

Step 3: Solve for n

Now simplify the equation to solve for \(n\). First, distribute the 4 on the left side to get \(4n + 4 = n + 2\). Then, subtract \(n\) from both sides to get \(3n + 4 = 2\). Lastly, subtract 4 from both sides and divide by 3 to get \(n = -\frac{2}{3}\)