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#mimetex(0.999...=0.\dot{9})
#mimetex(\frac{1}{3}\quad\quad\quad=0.\dot{3})
#mimetex(\frac{1}{3} * 3=0.\dot{3} * 3)
#mimetex(1\quad\quad\quad=0.\dot{9})
#mimetex(1-0.\dot{9}=0.\dot{0})
#mimetex(0.\dot{0} = 0)
#mimetex(0.\dot{9})
#mimetex(a_1=0.9)
#mimetex(r=0.1)
#mimetex(a_1r^{n-1})
#mimetex(S_n=\frac{a_1(r^n-1)}{r-1})
#mimetex(S_n=\frac{a_1(r^n-1)}{r-1}=\frac{0.9(0.1^n-1)}{-0.9})
#mimetex(n)
#mimetex(0.\dot{9}=\lim_{n\to\infty}S_n=\lim_{n\to\infty}\frac{0.9(0.1^n-1)}{-0.9}=\lim_{n\to\infty}-(0.1^n-1)=\lim_{n\to\infty}1-0.1^n=1)
#mimetex(0.\dot{9})
#mimetex(p\(1\)=\frac{99}{100})
#mimetex(p\(2\)=\frac{1}{100})
#mimetex(p\(3\)=\frac{1}{100}+\frac{1}{100}\cdot\frac{1}{99})
#mimetex(p\(3\)=\frac{1}{100}+\frac{1}{100}\cdot\frac{1}{99}=\frac{1}{100}\(1+\frac{1}{99}\)=\frac{1}{100}\(\frac{99}{99}+\frac{1}{99}\)=\frac{1}{100}\cdot\frac{100}{99}=\frac{1}{99})
#mimetex(p\(4\)=\frac{1}{100}+\frac{1}{100}\cdot\frac{1}{99}+\frac{1}{100}\cdot\frac{1}{98}+\frac{1}{100}\cdot\frac{1}{99}\cdot\frac{1}{98})
#mimetex(p\(3\)=\frac{1}{100}+\frac{1}{100}\cdot\frac{1}{99})
#mimetex(=\frac{1}{99})
#mimetex(p\(4\)=\frac{1}{100}+\frac{1}{100}\cdot\frac{1}{99}+\frac{1}{100}\cdot\frac{1}{98}+\frac{1}{100}\cdot\frac{1}{99}\cdot\frac{1}{98}=\frac{1}{99}+\frac{1}{99}\cdot\frac{1}{98})
#mimetex(=\frac{1}{99}\(1+\frac{1}{98}\)=\frac{1}{99}\cdot\frac{99}{98}=\frac{1}{98})
#mimetex(p\(n\)=\frac{1}{102-n})
#mimetex(p\(n\)=p\(n-1\)+p\(n-1\)\cdot\frac{1}{102-n})
#mimetex(p\(n\)=\frac{1}{102-n})
#mimetex(p\(n\)=p\(n-1\)+p\(n-1\)\cdot\frac{1}{102-n})
#mimetex(p\(2\)=\frac{1}{100})
#mimetex(p\(3\)=p\(2\)+p\(2\)\cdot\frac{1}{102-3}=\frac{1}{100}+\frac{1}{100}\cdot\frac{1}{99}=\frac{1}{99}=\frac{1}{102-n})
#mimetex(p\(n+1\)=p\(n\)+p\(n\)\cdot\frac{1}{102-\(n+1\)}=\frac{1}{102-n}+\frac{1}{102-n}\cdot\frac{1}{102-\(n+1\)}=\frac{1}{102-n}\(1+\frac{1}{102-n-1}\))
#mimetex(=\frac{1}{102-n}\(\frac{102-n-1}{102-n-1}+\frac{1}{102-n-1}\)=\frac{1}{102-n}\(\frac{102-n}{102-n-1}\)=\frac{1}{102-\(n+1\)})
#mimetex(p\(100\)=\frac{1}{102-100}=\frac{1}{2})
#mimetex(p\(n\))
#mimetex(p\(1\)=1)
#mimetex(p\(2\)=\frac{365}{365}\cdot\frac{364}{365})
#mimetex(p\(3\)=\frac{365}{365}\cdot\frac{364}{365}\cdot\frac{363}{365})
#mimetex(p\(n\)=\prod_{i=1}^{n}\frac{366-i}{365}=\frac{1}{365^n}\cdot\frac{365!}{(365-n)!})
#mimetex(1-\frac{1}{365^n}\cdot\frac{365!}{(365-n)!})
#mimetex(\frac{1}{4})
#mimetex(p\(n\))
#mimetex(f\(n\))
#mimetex(1-p=\(1-w^n\)^k);を書き換えて&mimetex(k = \frac{\log(1-p)}{\log(1-w^n)})
#mimetex(1-p=\(1-w^n\)^k)
#mimetex(k = \frac{\log(1-p)}{\log(1-w^n)})
#mimetex(\frac{\partial \bf{P}}{\partial \bf{t}})
#mimetex( -\frac{b \pm \sqrt{b^2-4ac}}{2a})
#mimetex( e^{i \pi} + 1 = \cos (\pi) + i\sin(\pi) + 1 = 0 )
#mimetex(D=(\bf{x} - \bf{\mu})^\top \Sigma^{-1} (\bf{x} - \bf{\mu}))
#mimetex(E=\sum_{k=1}^{n+m}p\(k\)k)
#mimetex(E=1 \times \frac{1}{2}+2\times \frac{1}{2}=1.5)
#mimetex(E=\sum_{k=1}^{m+1}k\frac{1}{m+1}=\frac{1}{m+1}\cdot \frac{1}{2}(m+1)(m+2)=\frac{1}{2}(m+2))
#mimetex(=\frac{1}{2}m+1)
#mimetex(E=\sum_{k=2}^{m+2}p\(k\)k=\frac{1}{10}2+\frac{2}{10}3+\frac{3}{10}4+\frac{4}{10}5=\frac{2+6+12+20}{10}=4)
#mimetex(E=\frac{1}{10}2+\frac{2}{10}3+\frac{3}{10}4+\frac{4}{10}5=\frac{1}{10}(1\cdot 2 + 2\cdot 3 + 3\cdot 4+4\cdot 5)=4)
#mimetex(M=\frac{(m+1)(m+2)}{2})
#mimetex(k\cdot (k-1))
#mimetex(E=\frac{1}{M}\sum_{k=2}^{m+2}k\cdot(k-1)=\frac{2}{(m+1)(m+2)}\sum_{k=2}^{m+2}k\cdot(k-1))
#mimetex(E=\frac{1}{M}\sum_{k=2}^{m+2}k\cdot(k-1)\\=\frac{2}{(m+1)(m+2)}\sum_{k=2}^{m+2}k^2-k=\frac{2}{(m+1)(m+2)}(\sum_{k=2}^{m+2}k^2-\sum_{k=2}^{m+2}k)\\=\frac{2}{(m+1)(m+2)}((\sum_{k=1}^{m+2}k^2-\sum_{k=1}^{1}k^2)-(\sum_{k=1}^{m+2}k-\sum_{k=1}^{1}k)))
#mimetex(\Sigma)
#mimetex(E=\frac{2}{(m+1)(m+2)}((\frac{1}{6}(m+2)(m+3)(2m+5)-1)-(\frac{1}{2}(m+2)(m+3)-1))\\=\frac{2}{(m+1)(m+2)}(\frac{1}{6}(m+2)(m+3)(2m+5)-\frac{1}{2}(m+2)(m+3)))
#mimetex(=\frac{1}{3(m+1)(m+2)}((m+2)(m+3)(2m+5)-3(m+2)(m+3))\\=\frac{(m+2)(m+3)(2m+2)}{3(m+1)(m+2)}=\frac{2(m+2)(m+3)(m+1)}{3(m+1)(m+2)}=\frac{2(m+3)}{3}\\E=\frac{2(m+3)}{3}=\frac{2m}{3}+2)
#mimetex(E=\frac{2m}{3}+2=\frac{2\cdot 3}{3}+2=2+2=4)
#mimetex(n=1)
#mimetex(\frac{1}{2}m+1)
#mimetex(n=2)
#mimetex(\frac{2}{3}m+2)
#mimetex(E=\frac{n}{n+1}m + n)
#mimetex(E=\sum_{k=n}^{n+m}p\(k\)k)
#mimetex([n,n+m])
#mimetex(p(k))
#mimetex(p(k))
#mimetex({}_{n+m} \mathrm{C}_{n})
#mimetex({}_{a+b} \mathrm{C}_{a})
#mimetex({}_{n+m+1} \mathrm{C}_{n+1})
#mimetex({}_{n+m-1}\mathrm{C}_{n-1})
#mimetex({}_{n+m}\mathrm{C}_{n})
#mimetex(p(k))
#mimetex(p(k)=\frac{{}_{k-1}\mathrm{C}_{n-1}}{{}_{n+m}\mathrm{C}_{n}})
#mimetex(E=\sum_{k=n}^{n+m}p\(k\)k\\=\sum_{k=n}^{n+m}\frac{{}_{k-1}\mathrm{C}_{n-1}}{{}_{n+m}\mathrm{C}_{n}}\cdot k)
#mimetex({}_a \mathrm{C}_b = \frac{a!}{(a-b)!b!})
#mimetex(E=\sum_{k=n}^{n+m}\frac{{}_{k-1}\mathrm{C}_{n-1}}{{}_{n+m}\mathrm{C}_{n}}\cdot k\\=\sum_{k=n}^{n+m}\frac{m!n!}{(n+m)!}\frac{(k-1)!}{(n-1)!(k-n)!}\cdot k\\=\frac{m!n!}{(n+m)!}\sum_{k=n}^{n+m}\frac{(k-1)!}{(n-1)!(k-n)!}\cdot k\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}\frac{(k-1)!}{(k-n)!}\cdot k)
#mimetex(\Sigma)
#mimetex((k-1)!k=k!)
#mimetex(\frac{k!}{(k-n)!}={}_k \mathrm{P}_n)
#mimetex(E=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}\frac{(k-1)!}{(k-n)!}\cdot k\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}\frac{k!}{(k-n)!}\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}{}_k\mathrm{P}_n)
#mimetex(\sum_{k=n}^{n+m}{}_k\mathrm{P}_n)
#mimetex(\sum_{k=1}^{a}{}_k \mathrm{P}_b=\frac{1}{(b+1)}\overbrace{(a+1)a(a-1)\cdots(a+1-b)}^{b+1})
#mimetex(E=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}{}_k\mathrm{P}_n\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}(\sum_{k=1}^{n+m}{}_k\mathrm{P}_n-\sum_{k=1}^{n-1}{}_k\mathrm{P}_n)\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}(\frac{1}{(n+1)}(n+m+1)(n+m)\cdots(m+1)-\frac{1}{(n+1)}n(n-1)\cdots(n-n))\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\frac{1}{(n+1)}\frac{(n+m+1)!}{m!}\\=\frac{m!n!}{(n-1)!m!}\frac{(n+m+1)!}{(n+m)!}\frac{1}{(n+1)}\\=\frac{n(n+m+1)}{(n+1)}=\frac{nm}{(n+1)}+\frac{n(n+1)}{(n+1)}=\frac{n}{n+1}m+n)
#mimetex(\frac{1}{(n+1)}n(n-1)\cdots(n-n))
#mimetex((n-n)=0)
#mimetex(E=\frac{n}{n+1}m + n)
#mimetex(\sum_{k=1}^{a}{}_k \mathrm{P}_b=\frac{1}{(b+1)}\overbrace{(a+1)a(a-1)\cdots(a+1-b)}^{b+1})
#mimetex(E=\frac{10}{11}m + 10)
#mimetex(p)
#mimetex(n)
#mimetex((1+n)^p -n^p -1)
#mimetex(p)
#mimetex((1+n)^p -n^p -1)
#mimetex(= ({}_pC_01^p + {}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1} + {}_pC_pn^p) \quad - n^p \quad - 1 )
#mimetex((1+n)^p -n^p -1)
#mimetex(= (1 + {}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1} + n^p) \quad - n^p \quad - 1 )
#mimetex(= {}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1})
#mimetex({}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1})
#mimetex(=\sum^{p-1}_{k=1}\left({}_pC_k1^{p-k}n^{k}\right));
#mimetex(=\sum^{p-1}_{k=1}\left({}_pC_kn^{k}\right))
#mimetex({}_pC_kn^{k}=pm)
#mimetex(m)
#mimetex(p)
#mimetex({}_pC_kn^{k})
#mimetex({}_pC_kn^{k});
#mimetex(=\frac{p!}{(p-k)!}\frac{1}{k!}n^k)
#mimetex(k \ge 1)
#mimetex(p)
#mimetex(k \ge 1)
#mimetex(p)
#mimetex({}_pC_kn^{k})
#mimetex(=pm)
#mimetex(m)
#mimetex((1+n)^p -n^p -1)
#mimetex(\begin{array}{crlllllll}p=1:&1&+n&&&&&-n&-1\\p=2:&1&+2n&+n^2&&&&-n^2&-1\\p=3:&1&+3n&+3n^2&+n^3&&&-n^3&-1\\p=4:&1&+4n&+6n^2&+4n^3&+n^4&&-n^4&-1\\p=5:&1&+5n&+10n^2&+10n^3&+5n^4&+n^5&-n^5&-1\\\end{array})
#mimetex(p=4)
#mimetex(=4)
#mimetex({}_pC_k);
#mimetex(=\frac{p!}{(p-k)!}\frac{1}{k!});
#mimetex(k)
#mimetex(0\le k \le)
#mimetex(p)
#mimetex(1\le k \le p-1)
#mimetex((1+n)^p -n^p -1)
#mimetex(=pm)
#mimetex((1+n)^p -n^p -1
#mimetex(p)
#mimetex(f(x)=\frac{4x+\sqrt{4x^2-1}}{\sqrt{2x+1}+\sqrt{2x-1}})
#mimetex(f(1)+f(2)+f(3)+\cdots +f(60)
#mimetex(a_n=\sqrt{2n-1})
#mimetex(a_{n+1}\cdot a_n=\sqrt{2n+1}\sqrt{2n-1}=\sqrt{4n^2-1})
#mimetex(a_{n+1}^2+a_n^2=2n+1+2n-1=4n)
#mimetex(a_{n+1}^2-a_n^2=2n+1-2n-1=2)
#mimetex(f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}=\frac{a_{n+1}^2+a_n^2+a_{n+1}\cdot a_n}{a_{n+1}+a_n}
#mimetex((a_{n+1}-a_n))
#mimetex(f(n)=\frac{(a_{n+1}-a_n)(a_{n+1}^2+a_n^2+a_{n+1}\cdot a_n)}{(a_{n+1}-a_n)(a_{n+1}+a_n)}=\frac{a_{n+1}^3+a_{n+1}^2a_n+a_{n+1}a_n^2-a_{n+1}^2a_n-a_{n+1}a_n^2-a_n^3}{a_{n+1}^2-a_n^2})
#mimetex(=\frac{a_{n+1}^3-a_n^3}{2}=-\frac{1}{2}a_n^3+\frac{1}{2}a_{n+1}^3)
#mimetex(f(n))
#mimetex(f(1)+f(2)+f(3)+\cdots +f(60))
#mimetex(=\left(-\frac{1}{2}a_1^3+\frac{1}{2}a_2^3\right)\quad +\left(-\frac{1}{2}a_2^3+\frac{1}{2}a_3^3\right)\quad +\left(-\frac{1}{2}a_3^3+\frac{1}{2}a_4^3\right)\quad +\cdots +\quad \left(-\frac{1}{2}a_{60}^3+\frac{1}{2}a_{61}^3\right))
#mimetex(=-\frac{1}{2}a_1^3+\frac{1}{2}a_{61}^3=-\frac{1}{2}\left(\sqrt{2-1}\right)^3+\left(\frac{1}{2}\sqrt{2\cdot 61 - 1}\right)^3)
#mimetex(=\frac{1}{2}(\sqrt{121}^3-1)=\frac{1}{2}(1331-1)=665)
#mimetex(p_1=1)
#mimetex(p_2=1)
#mimetex(p_{n+2}=p_{n+1}+p_n\left(n \ge 1\right))
#mimetex(\left{p_n\right})
#mimetex(p_n=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right\})
#mimetex(X_n\left(n=1,2,\cdots\right))
#mimetex(X_1=1)
#mimetex(X_n)
#mimetex(\alpha)
#mimetex(\alpha)
#mimetex(\alpha)
#mimetex(\alpha)
#mimetex(X_n)
#mimetex(X_{n+1})
#mimetex(X_1=1)
#mimetex(X_2=10)
#mimetex(X_3=101)
#mimetex(X_4=10110)
#mimetex(X_5=10110101)
#mimetex(X_n)
#mimetex(a_n)
#mimetex(X_n)
#mimetex(b_n)
#mimetex(b_1=0)
#mimetex(b_2=0)
#mimetex(b_3=1)
#mimetex(b_4=1)
#mimetex(b_5=3)
#mimetex(X_n)
#mimetex(I_n)
#mimetex(O_n)
#mimetex(a_n=I_n+O_n)
#mimetex(\left{\begin{array}{l}I_n=I_{n-1}+O_{n-1}\\O_n=I_{n-1}\end{array})
#mimetex(O_{n-1})
#mimetex(I_n)
#mimetex(\left{\begin{array}{l}I_n=I_{n-1}+I_{n-2}\\O_n=I_{n-1}\end{array})
#mimetex(I_n)
#mimetex(I_1=1)
#mimetex(I_2=1)
#mimetex(I_n=p_n=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right\})
#mimetex(O_n=I_{n-1}=p_{n-1}=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right\})
#mimetex(a_n)
#mimetex(a_n=I_n+O_n=p_n+p_{n-1}=p_{n+1})
#mimetex(a_n=p_{n+1}=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right\})
#mimetex(b_n)
#mimetex(p_n)
#mimetex(b_n)
#mimetex(p_n)
#mimetex(X_n)
#mimetex(X_n)
#mimetex(X_{n-1})
#mimetex(X_{n-2})
#mimetex(X_{n-1})
#mimetex(b_n=\left\{\begin{array}{lcl}b_{n-1}+b_{n-2}+1&:&n=odd\\b_{n-1}+b_{n-2}&:&else\end{array})
#mimetex((n\ge3))
#mimetex(c_n)
#mimetex(c_n=\left\{\begin{array}{ccl}1&:&n=odd\\0&:&else\end{array})
#mimetex(b_n)
#mimetex(b_n=b_{n-1}+b_{n-2}+c_n)
#mimetex((n\ge3))
#mimetex(b_n)
#mimetex(P_n)
#mimetex(b_n)
#mimetex(p_{n-1})
#mimetex(X_n)
#mimetex(b_n)
#mimetex(b_n=\left\{\begin{array}{lcl}p_{n-1}-1&:&n=even\\p_{n-1}&:&else\end{array})
#mimetex((n\ge2))
#mimetex(c_n)
#mimetex(b_n=p_{n-1}-1+c_n)
#mimetex((n\ge2))
#mimetex(d_n=1-c_n)
#mimetex(b_n=p_{n-1}-d_n)
#mimetex((n\ge2))
#mimetex(b_1=0)
#mimetex(b_2=p_1-d_2=1-(1-c_n)=1-1+0=0)
#mimetex(b_3=p_2-d_3=1-(1-c_n)=1-1+1=1)
#mimetex(b_n=p_{n-1}-d_n)
#mimetex(b_{n-1}=p_{n-2}-d_{n-1})
#mimetex(b_{n+1})
#mimetex(b_{n+1}=(b_n)+(b_{n-1})+c_{n+1}=(p_{n-1}-d_n)+(p_{n-2}-d_{n-1})+c_{n+1})
#mimetex(=(p_{n-1}+p_{n-2})+(-d_n-d_{n-1}+c_{n+1})=p_n+(-d_n-d_{n-1}+c_{n+1}))
#mimetex(-d_n-d_{n-1}=-1)
#mimetex(b_{n+1}=p_n+(-d_n-d_{n-1}+c_{n+1})=p_n+(-1+c_{n+1}))
#mimetex(d_n=1-c_n)
#mimetex(b_{n+1}=p_n+(-1+c_{n+1})=p_n+(-d_{n+1})=p_n-d_{n+1})
#mimetex(b_n=p_{n-1}-d_n)
#mimetex(n)
#mimetex(n+1)
#mimetex(b_n=p_{n-1}-d_n=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right\}-d_n)
#mimetex(b_n=p_{n-1}-d_n)
#mimetex((n\ge2));
#mimetex(E=\sum_{k=1}^{n+m}p\(k\)k)
#mimetex(E=1 \times \frac{1}{2}+2\times \frac{1}{2}=1.5)
#mimetex(E=\sum_{k=1}^{m+1}k\frac{1}{m+1}=\frac{1}{m+1}\cdot \frac{1}{2}(m+1)(m+2)=\frac{1}{2}(m+2))
#mimetex(=\frac{1}{2}m+1)
#mimetex(E=\sum_{k=2}^{m+2}p\(k\)k=\frac{1}{10}2+\frac{2}{10}3+\frac{3}{10}4+\frac{4}{10}5=\frac{2+6+12+20}{10}=4)
#mimetex(E=\frac{1}{10}2+\frac{2}{10}3+\frac{3}{10}4+\frac{4}{10}5=\frac{1}{10}(1\cdot 2 + 2\cdot 3 + 3\cdot 4+4\cdot 5)=4)
#mimetex(M=\frac{(m+1)(m+2)}{2})
#mimetex(k\cdot (k-1))
#mimetex(E=\frac{1}{M}\sum_{k=2}^{m+2}k\cdot(k-1)=\frac{2}{(m+1)(m+2)}\sum_{k=2}^{m+2}k\cdot(k-1))
#mimetex(E=\frac{1}{M}\sum_{k=2}^{m+2}k\cdot(k-1)\\=\frac{2}{(m+1)(m+2)}\sum_{k=2}^{m+2}k^2-k=\frac{2}{(m+1)(m+2)}(\sum_{k=2}^{m+2}k^2-\sum_{k=2}^{m+2}k)\\=\frac{2}{(m+1)(m+2)}((\sum_{k=1}^{m+2}k^2-\sum_{k=1}^{1}k^2)-(\sum_{k=1}^{m+2}k-\sum_{k=1}^{1}k)))
#mimetex(\Sigma)
#mimetex(E=\frac{2}{(m+1)(m+2)}((\frac{1}{6}(m+2)(m+3)(2m+5)-1)-(\frac{1}{2}(m+2)(m+3)-1))\\=\frac{2}{(m+1)(m+2)}(\frac{1}{6}(m+2)(m+3)(2m+5)-\frac{1}{2}(m+2)(m+3)))
#mimetex(=\frac{1}{3(m+1)(m+2)}((m+2)(m+3)(2m+5)-3(m+2)(m+3))\\=\frac{(m+2)(m+3)(2m+2)}{3(m+1)(m+2)}=\frac{2(m+2)(m+3)(m+1)}{3(m+1)(m+2)}=\frac{2(m+3)}{3}\\E=\frac{2(m+3)}{3}=\frac{2m}{3}+2)
#mimetex(E=\frac{2m}{3}+2=\frac{2\cdot 3}{3}+2=2+2=4)
#mimetex(n=1)
#mimetex(\frac{1}{2}m+1)
#mimetex(n=2)
#mimetex(\frac{2}{3}m+2)
#mimetex(E=\frac{n}{n+1}m + n)
#mimetex(E=\sum_{k=n}^{n+m}p\(k\)k)
#mimetex([n,n+m])
#mimetex(p(k))
#mimetex(p(k))
#mimetex({}_{n+m} \mathrm{C}_{n})
#mimetex({}_{a+b} \mathrm{C}_{a})
#mimetex({}_{n+m+1} \mathrm{C}_{n+1})
#mimetex({}_{n+m-1}\mathrm{C}_{n-1})
#mimetex({}_{n+m}\mathrm{C}_{n})
#mimetex(p(k))
#mimetex(p(k)=\frac{{}_{k-1}\mathrm{C}_{n-1}}{{}_{n+m}\mathrm{C}_{n}})
#mimetex(E=\sum_{k=n}^{n+m}p\(k\)k\\=\sum_{k=n}^{n+m}\frac{{}_{k-1}\mathrm{C}_{n-1}}{{}_{n+m}\mathrm{C}_{n}}\cdot k)
#mimetex({}_a \mathrm{C}_b = \frac{a!}{(a-b)!b!})
#mimetex(E=\sum_{k=n}^{n+m}\frac{{}_{k-1}\mathrm{C}_{n-1}}{{}_{n+m}\mathrm{C}_{n}}\cdot k\\=\sum_{k=n}^{n+m}\frac{m!n!}{(n+m)!}\frac{(k-1)!}{(n-1)!(k-n)!}\cdot k\\=\frac{m!n!}{(n+m)!}\sum_{k=n}^{n+m}\frac{(k-1)!}{(n-1)!(k-n)!}\cdot k\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}\frac{(k-1)!}{(k-n)!}\cdot k)
#mimetex(\Sigma)
#mimetex((k-1)!k=k!)
#mimetex(\frac{k!}{(k-n)!}={}_k \mathrm{P}_n)
#mimetex(E=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}\frac{(k-1)!}{(k-n)!}\cdot k\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}\frac{k!}{(k-n)!}\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}{}_k\mathrm{P}_n)
#mimetex(\sum_{k=n}^{n+m}{}_k\mathrm{P}_n)
#mimetex(\sum_{k=1}^{a}{}_k \mathrm{P}_b=\frac{1}{(b+1)}\overbrace{(a+1)a(a-1)\cdots(a+1-b)}^{b+1})
#mimetex(E=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\sum_{k=n}^{n+m}{}_k\mathrm{P}_n\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}(\sum_{k=1}^{n+m}{}_k\mathrm{P}_n-\sum_{k=1}^{n-1}{}_k\mathrm{P}_n)\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}(\frac{1}{(n+1)}(n+m+1)(n+m)\cdots(m+1)-\frac{1}{(n+1)}n(n-1)\cdots(n-n))\\=\frac{m!n!}{(n+m)!}\frac{1}{(n-1)!}\frac{1}{(n+1)}\frac{(n+m+1)!}{m!}\\=\frac{m!n!}{(n-1)!m!}\frac{(n+m+1)!}{(n+m)!}\frac{1}{(n+1)}\\=\frac{n(n+m+1)}{(n+1)}=\frac{nm}{(n+1)}+\frac{n(n+1)}{(n+1)}=\frac{n}{n+1}m+n)
#mimetex(\frac{1}{(n+1)}n(n-1)\cdots(n-n))
#mimetex((n-n)=0)
#mimetex(E=\frac{n}{n+1}m + n)
#mimetex(\sum_{k=1}^{a}{}_k \mathrm{P}_b=\frac{1}{(b+1)}\overbrace{(a+1)a(a-1)\cdots(a+1-b)}^{b+1})
#mimetex(E=\frac{10}{11}m + 10)
#mimetex(p)
#mimetex(n)
#mimetex((1+n)^p -n^p -1)
#mimetex(p)
#mimetex((1+n)^p -n^p -1)
#mimetex(= ({}_pC_01^p + {}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1} + {}_pC_pn^p) \quad - n^p \quad - 1 )
#mimetex((1+n)^p -n^p -1)
#mimetex(= (1 + {}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1} + n^p) \quad - n^p \quad - 1 )
#mimetex(= {}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1})
#mimetex({}_pC_1n + \ldots + {}_pC_{p-1}n^{p-1})
#mimetex(=\sum^{p-1}_{k=1}\left({}_pC_k1^{p-k}n^{k}\right));
#mimetex(=\sum^{p-1}_{k=1}\left({}_pC_kn^{k}\right))
#mimetex({}_pC_kn^{k}=pm)
#mimetex(m)
#mimetex(p)
#mimetex({}_pC_kn^{k})
#mimetex({}_pC_kn^{k});
#mimetex(=\frac{p!}{(p-k)!}\frac{1}{k!}n^k)
#mimetex(k \ge 1)
#mimetex(p)
#mimetex(k \ge 1)
#mimetex(p)
#mimetex({}_pC_kn^{k})
#mimetex(=pm)
#mimetex(m)
#mimetex((1+n)^p -n^p -1)
#mimetex(\begin{array}{crlllllll}p=1:&1&+n&&&&&-n&-1\\p=2:&1&+2n&+n^2&&&&-n^2&-1\\p=3:&1&+3n&+3n^2&+n^3&&&-n^3&-1\\p=4:&1&+4n&+6n^2&+4n^3&+n^4&&-n^4&-1\\p=5:&1&+5n&+10n^2&+10n^3&+5n^4&+n^5&-n^5&-1\\\end{array})
#mimetex(p=4)
#mimetex(=4)
#mimetex({}_pC_k);
#mimetex(=\frac{p!}{(p-k)!}\frac{1}{k!});
#mimetex(k)
#mimetex(0\le k \le)
#mimetex(p)
#mimetex(1\le k \le p-1)
#mimetex((1+n)^p -n^p -1)
#mimetex(=pm)
#mimetex((1+n)^p -n^p -1
#mimetex(p)
#mimetex(f(x)=\frac{4x+\sqrt{4x^2-1}}{\sqrt{2x+1}+\sqrt{2x-1}})
#mimetex(f(1)+f(2)+f(3)+\cdots +f(60)
#mimetex(a_n=\sqrt{2n-1})
#mimetex(a_{n+1}\cdot a_n=\sqrt{2n+1}\sqrt{2n-1}=\sqrt{4n^2-1})
#mimetex(a_{n+1}^2+a_n^2=2n+1+2n-1=4n)
#mimetex(a_{n+1}^2-a_n^2=2n+1-2n-1=2)
#mimetex(f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}=\frac{a_{n+1}^2+a_n^2+a_{n+1}\cdot a_n}{a_{n+1}+a_n}
#mimetex((a_{n+1}-a_n))
#mimetex(f(n)=\frac{(a_{n+1}-a_n)(a_{n+1}^2+a_n^2+a_{n+1}\cdot a_n)}{(a_{n+1}-a_n)(a_{n+1}+a_n)}=\frac{a_{n+1}^3+a_{n+1}^2a_n+a_{n+1}a_n^2-a_{n+1}^2a_n-a_{n+1}a_n^2-a_n^3}{a_{n+1}^2-a_n^2})
#mimetex(=\frac{a_{n+1}^3-a_n^3}{2}=-\frac{1}{2}a_n^3+\frac{1}{2}a_{n+1}^3)
#mimetex(f(n))
#mimetex(f(1)+f(2)+f(3)+\cdots +f(60))
#mimetex(=\left(-\frac{1}{2}a_1^3+\frac{1}{2}a_2^3\right)\quad +\left(-\frac{1}{2}a_2^3+\frac{1}{2}a_3^3\right)\quad +\left(-\frac{1}{2}a_3^3+\frac{1}{2}a_4^3\right)\quad +\cdots +\quad \left(-\frac{1}{2}a_{60}^3+\frac{1}{2}a_{61}^3\right))
#mimetex(=-\frac{1}{2}a_1^3+\frac{1}{2}a_{61}^3=-\frac{1}{2}\left(\sqrt{2-1}\right)^3+\left(\frac{1}{2}\sqrt{2\cdot 61 - 1}\right)^3)
#mimetex(=\frac{1}{2}(\sqrt{121}^3-1)=\frac{1}{2}(1331-1)=665)
#mimetex(p_1=1)
#mimetex(p_2=1)
#mimetex(p_{n+2}=p_{n+1}+p_n\left(n \ge 1\right))
#mimetex(\left{p_n\right})
#mimetex(p_n=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right\})
#mimetex(X_n\left(n=1,2,\cdots\right))
#mimetex(X_1=1)
#mimetex(X_n)
#mimetex(\alpha)
#mimetex(\alpha)
#mimetex(\alpha)
#mimetex(\alpha)
#mimetex(X_n)
#mimetex(X_{n+1})
#mimetex(X_1=1)
#mimetex(X_2=10)
#mimetex(X_3=101)
#mimetex(X_4=10110)
#mimetex(X_5=10110101)
#mimetex(X_n)
#mimetex(a_n)
#mimetex(X_n)
#mimetex(b_n)
#mimetex(b_1=0)
#mimetex(b_2=0)
#mimetex(b_3=1)
#mimetex(b_4=1)
#mimetex(b_5=3)
#mimetex(X_n)
#mimetex(I_n)
#mimetex(O_n)
#mimetex(a_n=I_n+O_n)
#mimetex(\left{\begin{array}{l}I_n=I_{n-1}+O_{n-1}\\O_n=I_{n-1}\end{array})
#mimetex(O_{n-1})
#mimetex(I_n)
#mimetex(\left{\begin{array}{l}I_n=I_{n-1}+I_{n-2}\\O_n=I_{n-1}\end{array})
#mimetex(I_n)
#mimetex(I_1=1)
#mimetex(I_2=1)
#mimetex(I_n=p_n=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right\})
#mimetex(O_n=I_{n-1}=p_{n-1}=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right\})
#mimetex(a_n)
#mimetex(a_n=I_n+O_n=p_n+p_{n-1}=p_{n+1})
#mimetex(a_n=p_{n+1}=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right\})
#mimetex(b_n)
#mimetex(p_n)
#mimetex(b_n)
#mimetex(p_n)
#mimetex(X_n)
#mimetex(X_n)
#mimetex(X_{n-1})
#mimetex(X_{n-2})
#mimetex(X_{n-1})
#mimetex(b_n=\left\{\begin{array}{lcl}b_{n-1}+b_{n-2}+1&:&n=odd\\b_{n-1}+b_{n-2}&:&else\end{array})
#mimetex((n\ge3))
#mimetex(c_n)
#mimetex(c_n=\left\{\begin{array}{ccl}1&:&n=odd\\0&:&else\end{array})
#mimetex(b_n)
#mimetex(b_n=b_{n-1}+b_{n-2}+c_n)
#mimetex((n\ge3))
#mimetex(b_n)
#mimetex(P_n)
#mimetex(b_n)
#mimetex(p_{n-1})
#mimetex(X_n)
#mimetex(b_n)
#mimetex(b_n=\left\{\begin{array}{lcl}p_{n-1}-1&:&n=even\\p_{n-1}&:&else\end{array})
#mimetex((n\ge2))
#mimetex(c_n)
#mimetex(b_n=p_{n-1}-1+c_n)
#mimetex((n\ge2))
#mimetex(d_n=1-c_n)
#mimetex(b_n=p_{n-1}-d_n)
#mimetex((n\ge2))
#mimetex(b_1=0)
#mimetex(b_2=p_1-d_2=1-(1-c_n)=1-1+0=0)
#mimetex(b_3=p_2-d_3=1-(1-c_n)=1-1+1=1)
#mimetex(b_n=p_{n-1}-d_n)
#mimetex(b_{n-1}=p_{n-2}-d_{n-1})
#mimetex(b_{n+1})
#mimetex(b_{n+1}=(b_n)+(b_{n-1})+c_{n+1}=(p_{n-1}-d_n)+(p_{n-2}-d_{n-1})+c_{n+1})
#mimetex(=(p_{n-1}+p_{n-2})+(-d_n-d_{n-1}+c_{n+1})=p_n+(-d_n-d_{n-1}+c_{n+1}))
#mimetex(-d_n-d_{n-1}=-1)
#mimetex(b_{n+1}=p_n+(-d_n-d_{n-1}+c_{n+1})=p_n+(-1+c_{n+1}))
#mimetex(d_n=1-c_n)
#mimetex(b_{n+1}=p_n+(-1+c_{n+1})=p_n+(-d_{n+1})=p_n-d_{n+1})
#mimetex(b_n=p_{n-1}-d_n)
#mimetex(n)
#mimetex(n+1)
#mimetex(b_n=p_{n-1}-d_n=\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right\}-d_n)
#mimetex(b_n=p_{n-1}-d_n)
#mimetex((n\ge2));



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