[tex]\bf S = 2^{1} + 2^{2} + 2^{3} +2^{4}+ ....+ 2^{2004}[/tex]
Grupăm câte 4 termeni și dăm factor comun pe 2 la puterea cea mai mică.
[tex]S = 2^{1}\cdot \Big( 2^{1-1} + 2^{2-1} + 2^{3-1} +2^{4-1} \Big)+...+ 2^{2001}\cdot \Big( 2^{0} + 2^{1} + 2^{2} +2^{3} \Big)[/tex]
[tex]S = 2^{1}\cdot \Big( 2^{0} + 2^{1} + 2^{2} +2^{3} \Big)+...+ 2^{2001}\cdot \Big( 2^{0} + 2^{1} + 2^{2} +2^{3} \Big)[/tex]
[tex]S = 2^{1}\cdot \Big( 1 + 2 + 4 +8\Big)+...+ 2^{2001}\cdot \Big( 1 + 2 + 4 +8 \Big)[/tex]
[tex]S = 2^{1}\cdot 15+...+ 2^{2001}\cdot 15[/tex]
[tex]S = 15\cdot \Big( 2^{1}+2^{5}+2^{9} +2^{13}+...+ 2^{2001}\Big)[/tex]
[tex]\pink{\boxed{~S = 3\cdot 5\cdot \Big(2^{1}+2^{5}+2^{9} +...+ 2^{2001}\Big)~\vdots~5~}}[/tex]
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[tex]\bf S = 2^{1} + 2^{2} + 2^{3} +2^{4}+ ....+ 2^{2004}[/tex]
Grupăm câte 3 termeni și dăm factor comun pe 2 la puterea cea mai mică.
[tex]S = 2^{1}\cdot \Big( 2^{1-1} + 2^{2-1} + 2^{3-1} \Big)+...+ 2^{2002}\cdot \Big( 2^{0} + 2^{1} + 2^{2} \Big)[/tex]
[tex]S = 2^{1}\cdot \Big( 2^{0} + 2^{1} + 2^{2} \Big)+...+ 2^{2002}\cdot \Big( 2^{0} + 2^{1} + 2^{2} \Big)[/tex]
[tex]S = 2^{1}\cdot \Big( 1 + 2 +4\Big)+...+ 2^{2002}\cdot \Big( 1 + 2 +4\Big)[/tex]
[tex]S = 2^{1}\cdot 7+...+ 2^{2002}\cdot 7[/tex]
[tex]\purple{\boxed{~S = 7\cdot \Big( 2^{1}+2^{4}+2^{8} +...+ 2^{2002}\Big)~\vdots~7~}}[/tex]
[tex]==pav38==[/tex]
