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Question Number 136098 by SOMEDAVONG last updated on 18/Mar/21

I).compute Σ_(k=0) ^(1000) C_(2000) ^(2k) .

I).compute1000k=0C20002k.

Answered by Olaf last updated on 18/Mar/21

(1−1)^(2000)  = 0 = Σ_(p=0) ^(2000) (−1)^p C_p ^(2000)   0 =  Σ_(k=0) ^(1000) (−1)^(2k) C_(2k) ^(2000) +Σ_(k=0) ^(999) (−1)^(2k+1) C_(2k+1) ^(2000)   0 =  Σ_(k=0) ^(1000) C_(2k) ^(2000) −Σ_(k=0) ^(999) C_(2k+1) ^(2000)   Σ_(k=0) ^(1000) C_(2k) ^(2000)  = Σ_(k=0) ^(999) C_(2k+1) ^(2000)     (1+1)^(2000)  = 2^(2000)  =  Σ_(p=0) ^(2000) C_p ^(2000)   2^(2000)  =  Σ_(k=0) ^(1000) C_(2k) ^(2000) +Σ_(k=0) ^(999) C_(2k+1) ^(2000)   2^(2000)  =  Σ_(k=0) ^(1000) C_(2k) ^(2000) +Σ_(k=0) ^(1000) C_(2k) ^(2000)   2^(2000)  =  2Σ_(k=0) ^(1000) C_(2k) ^(2000)   Σ_(k=0) ^(1000) C_(2k) ^(2000)  = (1/2)(2^(2000) ) = 2^(1999)

(11)2000=0=2000p=0(1)pCp20000=1000k=0(1)2kC2k2000+999k=0(1)2k+1C2k+120000=1000k=0C2k2000999k=0C2k+120001000k=0C2k2000=999k=0C2k+12000(1+1)2000=22000=2000p=0Cp200022000=1000k=0C2k2000+999k=0C2k+1200022000=1000k=0C2k2000+1000k=0C2k200022000=21000k=0C2k20001000k=0C2k2000=12(22000)=21999

Commented by SOMEDAVONG last updated on 18/Mar/21

  THANKS   TEACHER

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