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Question Number 113796 by aurpeyz last updated on 15/Sep/20
find the greatest coeeficient in the expansion   (6−4x)^(−3)
findthegreatestcoeeficientintheexpansion(64x)3
Answered by mr W last updated on 15/Sep/20
(6−4x)^(−3) =6^(−3) (1−(2/3)x)^(−3)   =6^(−3) Σ_(k=0) ^∞ C_2 ^(k+2) ((2/3))^k x^k   let C_2 ^(k+2) ((2/3))^k ≥C_2 ^(k+3) ((2/3))^(k+1)   (((k+2)(k+1))/2)≥(((k+3)(k+2))/2)((2/3))  3(k+1)≥2(k+3)  k≥3  max. coef. is at k=3:  6^(−3) ×C_2 ^5 ×((2/3))^3 =((10)/(729))
(64x)3=63(123x)3=63k=0C2k+2(23)kxkletC2k+2(23)kC2k+3(23)k+1(k+2)(k+1)2(k+3)(k+2)2(23)3(k+1)2(k+3)k3max.coef.isatk=3:63×C25×(23)3=10729
Commented by aurpeyz last updated on 16/Sep/20
pls explain line 2 and 3 in words. thanks alot  sir
plsexplainline2and3inwords.thanksalotsir
Commented by mr W last updated on 16/Sep/20
line 2 is just binomial theorem
line2isjustbinomialtheorem
Commented by mr W last updated on 16/Sep/20
Commented by mr W last updated on 16/Sep/20
if A_n  is the largest, it means:  A_1 <A_2 <...<A_(n−1) <A_n >A_(n+1) >A_(n+2) >...  i.e. n is the minimum of k which  fulfills A_k ≥A_(k+1)   or C_2 ^(k+2) ((2/3))^k ≥C_2 ^(k+3) ((2/3))^(k+1)   ⇒k≥3  it means A_3  is the largest.
ifAnisthelargest,itmeans:A1<A2<<An1<An>An+1>An+2>i.e.nistheminimumofkwhichfulfillsAkAk+1orC2k+2(23)kC2k+3(23)k+1k3itmeansA3isthelargest.
Commented by aurpeyz last updated on 16/Sep/20
i understand that the general term of a  binomial is Σ_(r=o) ^n C_r ^n x^(n−r) y^r . pls explain this ⇒  Σ_(k=0) ^∞ C_2 ^(k+2) ((2/3))^k x^k . how you get to use k+2.
iunderstandthatthegeneraltermofabinomialisnr=oCnrxnryr.plsexplainthisk=0C2k+2(23)kxk.howyougettousek+2.
Commented by aurpeyz last updated on 16/Sep/20
so i that i will be able this understanding to   solve other problems. thanks alot
soithatiwillbeablethisunderstandingtosolveotherproblems.thanksalot
Commented by mr W last updated on 16/Sep/20
you should study the case (x+y)^n   with n=negative!
youshouldstudythecase(x+y)nwithn=negative!
Commented by aurpeyz last updated on 16/Sep/20
Commented by mr W last updated on 16/Sep/20
generally C_r ^n =C_(n−r) ^n   ⇒C_k ^(k+2) =C_2 ^(k+2)
generallyCrn=CnrnCkk+2=C2k+2
Commented by aurpeyz last updated on 16/Sep/20
wow. thanks Sir. i really appreciate. perfect
wow.thanksSir.ireallyappreciate.perfect

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