Fie \( n\in N \ \ n\ge 2 \). Aratati ca urmatoarele afirmatii sunt echivalente:
a) exista n numere naturale impare \( a_1,a_2,...,a_n \) astfel incat
\( a_1+a_2+ ...+a_n=a_1 a_2...a_n \)
b)\( 4|n-1 \).
n numere impare
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"\( \Longrightarrow \)"
presupunem ca sunt p numere de forma {4k+1} si q de forma {4k+3}, p+q=n.
not P=\( a_1a_2...a_n \)
S=\( a_1+a_2+...+a_n \)
pt q=M4, P=M4+1 => p=M4+1
pt q=M4+2, P=M4+1 => p=M4+3
pt q=M4+1, P=M4+3 => p=M4
pt q=M4+3, P=M4+3 => p=M4+2
"\( \Longleftarrow \)"
fie \( a_1=a_2=...+a_{n-2}=1 \)
\( a_{n-1}=3 \)
\( a_{n}=\frac{n+1}{2} \)
presupunem ca sunt p numere de forma {4k+1} si q de forma {4k+3}, p+q=n.
not P=\( a_1a_2...a_n \)
S=\( a_1+a_2+...+a_n \)
pt q=M4, P=M4+1 => p=M4+1
pt q=M4+2, P=M4+1 => p=M4+3
pt q=M4+1, P=M4+3 => p=M4
pt q=M4+3, P=M4+3 => p=M4+2
"\( \Longleftarrow \)"
fie \( a_1=a_2=...+a_{n-2}=1 \)
\( a_{n-1}=3 \)
\( a_{n}=\frac{n+1}{2} \)