Determinati ultima cifra a numarului \( n \) , unde :
\( n = 1 + 1 + 2 +2^2 +2^3 + ... + 2^{2004} \)
Ultima cifra a lui N
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miruna.lazar
- Bernoulli
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Folosim: \( 2^n+2^n=2^n(1+1)=2^n\cdot2^1=2^{n+1} \)
\( 2+2+2^2+2^3+...+2^{2004}=2^2+2^2+2^3+...+2^{2004}=...=2^{2004}+2^{2004}=2^{2005} \)
Aflam ultima cifra a lui \( 2^{2005} \).
\( 2^4=16 \Rightarrow 2^{4k}=\overline{.....6} \) (cand exponentul e divizibil prin 4 ultima cifra e intotdeauna 6)
\( 2^{2005}=2^{501\cdot4}\cdot2=\overline{......6}\cdot2=\overline{......2} \)
\( 2+2+2^2+2^3+...+2^{2004}=2^2+2^2+2^3+...+2^{2004}=...=2^{2004}+2^{2004}=2^{2005} \)
Aflam ultima cifra a lui \( 2^{2005} \).
\( 2^4=16 \Rightarrow 2^{4k}=\overline{.....6} \) (cand exponentul e divizibil prin 4 ultima cifra e intotdeauna 6)
\( 2^{2005}=2^{501\cdot4}\cdot2=\overline{......6}\cdot2=\overline{......2} \)