$b_n=a_1+a_2+a_3+.......+a_{n-1}+a_n$ olsun
$m_i$ ve $k_i$ dizi olsunlar;
$m_i=m_1,m_2,m_3,m_4,.........m_{n-1},m_n$
$k_i=k_1,k_2,k_3,k_4,.........k_{n-1},k_n$
Amacım genel gösterimi vermek , birazda olsa $k_i$ ve $m_i$ ler hakkında fıkır zemini oluşturmak.
Tek tek bazı açılımlar yapalımki genel gösterime tam uysunlar....
ana kural
$(x+y)^n=\large\displaystyle\sum\limits_{k=0}^{n}\dbinom{n}{k}.x^k.y^{n-k}$
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$b_{n}^{m_{n}}=[b_{n-1}+a_{n}]^{m_{n}}=\large\displaystyle\sum\limits_{k_{n}=0}^{m_{n}}\dbinom{m_{n}}{k_{n}}b_{(n-1)}^{k_{n}}a_{n}^{(m_{n}-k_{n})}$
$b_{(n-1)}^{k_{n}}=[b_{n-2}+a_{(n-1)}]^{k_{n}}=\large\displaystyle\sum\limits_{m_{n-1}=0}^{k_{n}}\dbinom{k_{n}}{m_{n-1}}b_{(n-2)}^{m_{(n-1)}}a_{(n-1)}^{(k_{n}-m_{(n-1)})}$
$b_{n-2}^{m_{n-1}}=[b_{n-3}+a_{n-2}]^{m_{n-1}}=\large\displaystyle\sum\limits_{k_{n-1}=0}^{m_{n-1}}\dbinom{m_{n-1}}{k_{n-1}}b_{(n-3)}^{k_{n-1}}a_{n-2}^{(m_{n-1}-k_{n-1})}$
$b_{(n-3)}^{k_{n-1}}=[b_{n-4}+a_{(n-3)}]^{k_{n-1}}=\large\displaystyle\sum\limits_{m_{n-2}=0}^{k_{n-1}}\dbinom{k_{n-1}}{m_{n-2}}b_{(n-4)}^{m_{(n-2)}}a_{(n-3)}^{(k_{n-1}-m_{(n-2)})}$
$b_{n-4}^{m_{n-2}}=[b_{n-5}+a_{n-4}]^{m_{n-2}}=\large\displaystyle\sum\limits_{k_{n-2}=0}^{m_{n-2}}\dbinom{m_{n-2}}{k_{n-2}}b_{(n-5)}^{k_{n-2}}a_{n-4}^{(m_{n-2}-k_{n-2})}$
$b_{(n-5)}^{k_{n-2}}=[b_{n-6}+a_{(n-5)}]^{k_{n-2}}=\large\displaystyle\sum\limits_{m_{n-3}=0}^{k_{n-2}}\dbinom{k_{n-2}}{m_{n-3}}b_{(n-6)}^{m_{(n-3)}}a_{(n-5)}^{(k_{n-2}-m_{(n-3)})}$
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yani
Her birini bir öncekinde yerine koyarsak.....
$b_{n}^{m_{n}}=[b_{n-1}+a_{n}]^{m_{n}}=\large\displaystyle\sum\limits_{k_{n}=0}^{m_{n}}\dbinom{m_{n}}{k_{n}}\left[\large\displaystyle\sum\limits_{m_{n-1}=0}^{k_{n}}\dbinom{k_{n}}{m_{n-1}}\left[ \large\displaystyle\sum\limits_{k_{n-1}=0}^{m_{n-1}}\dbinom{m_{n-1}}{k_{n-1}}\left[\large\displaystyle\sum\limits_{m_{n-2}=0}^{k_{n-1}}\dbinom{k_{n-1}}{m_{n-2}}\left[\ddots_{\ddots_{\ddots_{\ddots}}}\right]a_{(n-3)}^{(k_{n-1}-m_{(n-2)})}\right]a_{n-2}^{(m_{n-1}-k_{n-1})} \right]a_{(n-1)}^{(k_{n}-m_{(n-1)})}\right]a_{n}^{(m_{n}-k_{n})}$