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Data publicarii: 26 Octombrie, 2014

TEORIE

(a+b)^n=C_n^0a^nb^0+C_n^1a^{n-1}b^1+C_n^2a^{n-2}b^2+\cdots+C_n^ka^{n-k}b^k+\cdots+C_n^na^0b^n=(a+b)^n=C_n^0a^nb^0+C_n^1a^{n-1}b^1+C_n^2a^{n-2}b^2+\cdots+C_n^ka^{n-k}b^k+\cdots+C_n^na^0b^n= \sum_{k=0}^{k=n}{C_n^ka^{n-k}b^k};\sum_{k=0}^{k=n}{C_n^ka^{n-k}b^k};  

T_{k+1}=C_n^ka^{n-k}b^k;(termenul\;general)T_{k+1}=C_n^ka^{n-k}b^k;(termenul\;general)

Cazuri particulare: 

n=2 => (a ± b)² = a² ± 2ab + b;

n=3 => (a ± b)³ = a³ ± 3a²b + 3ab² ± b³;

a=b=1\Rightarrow{\sum_{k=0}^{k=n}{C_n^k}=C_n^0+C_n^1+C_n^2+\cdots+C_n^k+\cdots+C_n^n=2^n};a=b=1\Rightarrow{\sum_{k=0}^{k=n}{C_n^k}=C_n^0+C_n^1+C_n^2+\cdots+C_n^k+\cdots+C_n^n=2^n};

Consecinte: 

C_n^0+C_n^2+C_n^4+\cdots=C_n^1+C_n^3+C_n^5+\cdots=2^{n-1};C_n^0+C_n^2+C_n^4+\cdots=C_n^1+C_n^3+C_n^5+\cdots=2^{n-1};   

\sum_{k=0}^{k=n}{(-1)^k}\cdot{C_n^k}=C_n^0-C_n^1+C_n^2-C_n^3+\cdots+(-1)^n\cdot{C_n^n}=0\sum_{k=0}^{k=n}{(-1)^k}\cdot{C_n^k}=C_n^0-C_n^1+C_n^2-C_n^3+\cdots+(-1)^n\cdot{C_n^n}=0  


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maria teodorescu, 21.03.2017 13:41

f bun

 

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