How can the inequality problem be solved for 1.8^n/n! < 0.201?

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In summary, The problem asks to solve the inequality 1.8^n/n! < 0.201 and the suggested approach is to use Stirling's approximation for factorial, n! ≈ √(2πn)(n/e)^n, for large n. However, it is unclear how large n needs to be and the possibility of using brute force to test n values is also mentioned.
  • #1
thereddevils
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Homework Statement



Solve the inequality,

[tex]\frac{1.8^n}{n!}<0.201[/tex]


Homework Equations





The Attempt at a Solution



some hints?
 
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  • #2


have you considered Stirlings approximation for the factorial?
 
  • #3


for n>>1
[tex] n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n [/tex]
 
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  • #4


though you'll have to check that n is pretty big... clearly n>8 here but how big...?
 
  • #5


n=6, is 6 considered large?
 
  • #6


no sorry, i though it was 8^n...

can you just use brute force then & test n values?
 
  • #7


lanedance said:
no sorry, i though it was 8^n...

can you just use brute force then & test n values?

yeah, that's how i got n=6 but i just wonder is there a tidier and definite way of solving that.
Thanks for helping me thus far.
 

FAQ: How can the inequality problem be solved for 1.8^n/n! < 0.201?

What is an inequality problem?

An inequality problem is a mathematical statement that compares two quantities using an inequality symbol, such as <, >, ≤, or ≥. The goal is to find the possible values that satisfy the inequality, rather than just one specific solution.

How do I solve an inequality problem?

To solve an inequality problem, you need to follow the same rules as solving an equation, such as combining like terms, using inverse operations, and isolating the variable. However, when you multiply or divide both sides by a negative number, the inequality symbol must be flipped.

What is the difference between an open and closed circle in an inequality graph?

In an inequality graph, an open circle (○) indicates that the value is not included in the solution, while a closed circle (●) indicates that the value is included in the solution. For example, in the inequality x > 2, the graph would have an open circle at 2, because 2 is not a possible solution.

Can I check my answer to an inequality problem?

Yes, you can always check your answer to an inequality problem by plugging it back into the original inequality and seeing if it satisfies the statement. If it does, then it is a valid solution.

What are some common mistakes to avoid when solving an inequality problem?

One common mistake is forgetting to flip the inequality symbol when multiplying or dividing by a negative number. Another mistake is treating an inequality problem like an equation and only finding one specific solution, rather than the range of possible solutions. It is important to also pay attention to the direction of the inequality symbol and any restrictions on the variable, such as x > 0.

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