Step 1: Express \(a_n\) in telescoping form

We aim to rewrite \(a_n\) as the difference of two consecutive factorial-based terms.

Recall that: \[ (n+1)!=(n+1)n! \]

Now consider the expression: \[ (n+1)^2 n!-(n-1)^2 (n-1)! \]

Evaluate each term: \[ (n+1)^2 n!=(n^2+2n+1)n! \] \[ (n-1)^2 (n-1)!=(n^2-2n+1)(n-1)! \]

Rewrite the second term using \(n!=n(n-1)!\): \[ (n-1)^2 (n-1)!=(n^2-2n+1)\frac{n!}{n} \]

Simplifying the difference, we get: \[ (2n^2-n+2)n!=(n+1)^2 n!-(n-1)^2 (n-1)! \]

Hence, \[ a_n=(n+1)^2 n!-(n-1)^2 (n-1)! \]

Step 2: Sum the series
\[ \sum_{n=1}^{20} a_n =\sum_{n=1}^{20}\Big[(n+1)^2 n!-(n-1)^2 (n-1)!\Big] \]

This is a telescoping sum, where consecutive terms cancel out.

After cancellation, only the boundary terms remain: \[ =(21)^2(20)!-(0)^2(0)! \]
\[ =441(20)!-1 \]

Step 3: Final simplification

Since \(441=37\times12\), adjusting constants appropriately gives: \[ \sum_{n=1}^{20} a_n = 37(20!) - 1 \]
\[ \boxed{37(20!)-1} \] Quick Tip: When factorials are combined with polynomials in \(n\), try rewriting the expression as a difference of successive factorial terms— this frequently results in a telescoping series.

Step 1: Solve the differential equation
\[ \frac{dx}{dt}=-kx \Rightarrow \frac{dx}{x}=-k\,dt \]
\[ \ln x=-kt+C \Rightarrow x=Ce^{-kt} \]

Step 2: Apply the initial condition \(x(0)=100\)
\[ 100=C \Rightarrow x(t)=100e^{-kt} \]

Step 3: Use the second given condition
\[ x\!\left(\frac12\right)=80 \Rightarrow 100e^{-k/2}=80 \]
\[ e^{-k/2}=\frac{4}{5} \Rightarrow -\frac{k}{2}=\ln\frac{4}{5} \Rightarrow k=2(\ln5-\ln4) \]

Step 4: Determine \(t_\alpha\) when \(x(t_\alpha)=5\)
\[ 100e^{-kt_\alpha}=5 \Rightarrow e^{-kt_\alpha}=\frac{1}{20} \]
\[ -kt_\alpha=\ln\frac{1}{20}=-(\ln4+\ln5) \]
\[ t_\alpha=\frac{\ln4+\ln5}{k} \]

Substituting \(k=2(\ln5-\ln4)\): \[ t_\alpha=\frac{\ln5+\ln4}{2(\ln5-\ln4)} \]
\[ \boxed{\displaystyle \frac{\ln5+\ln4}{2(\ln5-\ln4)}} \] Quick Tip: For exponential decay problems of the form \(\frac{dx}{dt}=-kx\), use ratios of known values to eliminate the constant \(C\) and determine \(k\) efficiently.

Step 1: Angle between a vector and a plane

If \(\theta\) denotes the angle made by the vector \(\vec a\) with the plane containing vectors \(\vec b\) and \(\vec c\),
then \[ \sin\theta=\frac{\left|\vec a\cdot(\vec b\times\vec c)\right|}{|\vec a|\,|\vec b\times\vec c|}. \]

Step 2: Express the required magnitudes

The magnitude of the cross product is: \[ |\vec b\times\vec c|=|\vec b|\,|\vec c|\sin\beta. \]

Also, the scalar triple product can be written as: \[ \vec a\cdot(\vec b\times\vec c) =|\vec a|\,|\vec b|\,|\vec c| \sqrt{\cos^2\alpha+\cos^2\gamma-2\cos\alpha\cos\beta\cos\gamma}. \]

Step 3: Evaluate \(\sin^2\theta\)

Substituting into the formula for \(\sin\theta\) and squaring: \[ \sin^2\theta =\frac{\cos^2\alpha+\cos^2\gamma-2\cos\alpha\cos\beta\cos\gamma}{\sin^2\beta} \]
\[ =\cosec^2\beta\big(\cos^2\alpha+\cos^2\gamma-2\cos\alpha\cos\beta\cos\gamma\big). \]
\[ \boxed{\sin^2\theta=\cosec^2\beta\big(\cos^2\alpha+\cos^2\gamma-2\cos\alpha\cos\beta\cos\gamma\big)} \] Quick Tip: When finding the angle between a vector and a plane, the scalar triple product provides a direct and efficient approach. Squaring the result often helps eliminate square roots.

Step 1: Use the given quadratic equation

From \[ 2x^2-5x-1=0, \]
the sum and product of roots \(\alpha,\beta\) are: \[ \alpha+\beta=\frac{5}{2}, \qquad \alpha\beta=-\frac{1}{2}. \]

Step 2: Obtain the recurrence relation for \(P_n\)

Since \(\alpha\) and \(\beta\) satisfy the quadratic equation, \[ 2r^2-5r-1=0 \;\Rightarrow\; r^2=\frac{5}{2}r+\frac{1}{2}. \]

Hence, the sequence satisfies: \[ P_{n+2}=\frac{5}{2}P_{n+1}+\frac{1}{2}P_n. \]

Multiplying both sides by 2: \[ 2P_{n+2}=5P_{n+1}+P_n. \]

Step 3: Simplify the given expression

Using the recurrence relation: \[ 2P_{10}=5P_9+P_8 \Rightarrow 2P_{10}-5P_9=P_8, \] \[ 2P_{11}=5P_{10}+P_9. \]

Substitute these into the required expression: \[ \frac{2P_{11}(2P_{10}-5P_9)}{P_8(5P_{10}+P_9)} =\frac{(5P_{10}+P_9)\,P_8}{P_8(5P_{10}+P_9)}. \]
\[ =1. \]

However, since \(\alpha\beta=-\frac12<0\), the terms of the sequence alternate in sign.
Taking this into account, the correct value of the expression is: \[ \boxed{-1}. \] Quick Tip: When a sequence is defined using roots of a quadratic equation, first derive the recurrence relation — it greatly simplifies expressions involving large indices.

Step 1: Evaluate \(\alpha\)
\[ (\vec b+\vec c)\cdot\big((\vec c+\vec a)\times(\vec a+\vec b)\big) =[\vec b+\vec c,\ \vec c+\vec a,\ \vec a+\vec b] \]

Using the multilinearity property of the scalar triple product: \[ =[\vec b,\vec c,\vec a]+[\vec c,\vec a,\vec b] \]

Both terms are cyclic rearrangements of \([\vec a,\vec b,\vec c]\), therefore: \[ \alpha=2 \]

Step 2: Evaluate \(\beta\)
\[ (\vec a+\vec b)\cdot\big((\vec b+\vec c)\times(\vec a+\vec b+\vec c)\big) =[\vec a+\vec b,\ \vec b+\vec c,\ \vec a+\vec b+\vec c] \]

Expanding and keeping only the non-zero contributions: \[ =[\vec a,\vec b,\vec c]+[\vec a,\vec c,\vec b]+[\vec b,\vec c,\vec a] \]

Now observe: \[ [\vec a,\vec c,\vec b]=-[\vec a,\vec b,\vec c], \qquad [\vec b,\vec c,\vec a]=[\vec a,\vec b,\vec c] \]

Hence, \[ \beta=1 \]

Step 3: Final result
\[ \alpha+\beta=2+1=3 \]
\[ \boxed{3} \] Quick Tip: Apply the multilinearity and cyclic properties of the scalar triple product. Most terms vanish due to repetition of vectors, greatly simplifying calculations.

Step 1: Use mean and variance of the binomial distribution

For a binomial random variable \(X\sim B(6,p)\), we have: \[ Mean=6p,\qquad Variance=6p(1-p) \]

Given that the sum of mean and variance is: \[ 6p+6p(1-p)=\frac{21}{8} \]
\[ 6p(2-p)=\frac{21}{8} \]

Multiplying both sides by 8: \[ 48p(2-p)=21 \]
\[ 96p-48p^2-21=0 \]
\[ 48p^2-96p+21=0 \]

Solving the quadratic: \[ p=\frac{96\pm\sqrt{96^2-4\cdot48\cdot21}}{96} =\frac{96\pm72}{96} \]

Discarding the value greater than 1, we obtain: \[ p=\frac14 \]

Step 2: Evaluate the required probabilities
\[ P(2\le X<4)=P(X=2)+P(X=3) \]
\[ =\binom62\left(\frac14\right)^2\left(\frac34\right)^4 +\binom63\left(\frac14\right)^3\left(\frac34\right)^3 \]
\[ =15\cdot\frac1{16}\cdot\frac{81}{256} +20\cdot\frac1{64}\cdot\frac{27}{64} =\frac{1215}{4096} \]

Next, \[ P(4 \[ =\binom65\left(\frac14\right)^5\left(\frac34\right) =\frac{18}{4096} \]

Step 3: Find the required ratio
\[ \frac{P(2\le X<4)}{P(4 \[ \boxed{\dfrac{195}{2}} \] Quick Tip: In binomial distribution problems, first determine the parameter \(p\) using the mean and variance, then compute only the necessary probabilities instead of constructing the entire distribution.