If the value of sum of first 5 non-zero natural numbers is equal to 6!𝑥446!x​ , then find the value of 1𝑥x1​ ?

Break Down the Problem

1. Identify the sum of the first 5 non-zero natural numbers.
2. Evaluate $6!$ and $446!$.
3. Set up the equation based on the given information and solve for $x$.
4. Find the value of $\frac{1}{x}$.

Relevant Concepts

• The sum of the first $n$ natural numbers is given by the formula: $S = \frac{n(n + 1)}{2}$
• Factorial is denoted as $n!$, which is the product of all positive integers up to $n$.

Analysis and Detail

1. Sum Calculation:

   The sum of the first 5 non-zero natural numbers: $S = 1 + 2 + 3 + 4 + 5 = 15$

2. Factorial Calculation:

   Compute $6!$: $6! = 720$

   The equation given is: $15 = 6! \cdot x \cdot 446!$ Therefore, substituting $6!$: $15 = 720 \cdot x \cdot 446!$

3. Solve for $x$:

   Rearranging the equation gives: $x = \frac{15}{720 \cdot 446!}$

Verify and Summarize

The calculation for $x$ was derived directly from the summation and factorial. We verified the sum calculation and factorial definitions were correct.

Final Answer

To find $\frac{1}{x}$: $\frac{1}{x} = \frac{720 \cdot 446!}{15} = 48 \cdot 446!$