Question Number 102278 by I want to learn more last updated on 08/Jul/20

Find the greatest coefficient in the following without  actually expand.  (i)        (5  −  3x)^(10)   (ii)        (5  +  3x)^(− 10)

Commented bymr W last updated on 08/Jul/20

see Q94397

Commented bymr W last updated on 08/Jul/20

see Q91464

Commented byI want to learn more last updated on 08/Jul/20

Sir am going say    5^(10) Σ_(k  =  0) ^(10)   ^(10) C_k  (− ((3x)/5))^k   Sir am going say    5^(10) Σ_(k  =  0) ^(10)   ^(10) C_k  (− (3/5))^k .x^k   Am concern about the minus.    And here,         (3  +  2x)^(− 7)    Sir am going say   (1/(3^7  )) Σ_(k  =  0) ^∞   ^(k  +  6) C_k  (− ((2x)/3))^k     The signs is my concern.  Just like the opposite sign to the one you solved sir.

Commented bymr W last updated on 08/Jul/20

i showed you two different ways how  to treat the ♮−ε sign.

Answered by john santu last updated on 08/Jul/20

(1)C_5 ^(10)  5^5  (−3x)^6  = ((10.9.8.6)/(5.4.3.2.1)). 5^5 .(−3)^6   = 5^5 . 72. 3^6

Commented byI want to learn more last updated on 08/Jul/20

Thanks sir

Answered by mr W last updated on 08/Jul/20

(i)  (5−3x)^(10) =Σ_(k=0) ^(10) C_k ^(10) 5^(10−k) (−3x)^k   =Σ_(k=0) ^(10) (−1)^k C_k ^(10) 3^k 5^(10−k) x^k   a_k =(−1)^k C_k ^(10) 3^k 5^(10−k)   for greatest coefficient a_k  we should  only look at the positive values, i.e.  if k is even: k=2n  a_(2n) =C_(2n) ^(10) 3^(2n) 5^(10−2n)   =5^(10) C_(2n) ^(10) ((9/(25)))^n   let C_(2n) ^(10) ((9/(25)))^n >C_(2(n+1)) ^(10) ((9/(25)))^(n+1)   ((10!)/((2n)!(10−2n)!))((9/(25)))^n >((10!)/((2n+2)!(10−2n−2)!))((9/(25)))^(n+1)   (1/((10−2n)(10−2n−1)))>(1/((2n+2)(2n+1)))×(9/(25))  ((25)/(45−19n+2n^2 ))>(9/(2n^2 +3n+1))  16n^2 +123n−190>0  n>((−123+(√(123^2 +4×16×190)))/(32))≈1.3  ⇒n=2  i.e. a_4  is the greates coefficient.  a_4 =5^(10) ×C_4 ^(10) ×((9/(25)))^2 =265 781 250

Commented byI want to learn more last updated on 09/Jul/20

I really appreciate sir

Answered by mr W last updated on 08/Jul/20

(ii)  (5+3x)^(−10)   =(1/5^(10) )(1+((3x)/5))^(−10)   =(1/5^(10) )Σ_(k=0) ^∞ C_9 ^(k+9) (−(3/5))^k x^k   let C_9 ^(k+9) ((3/5))^k >C_9 ^(k+10) ((3/5))^(k+1)   (((k+9)!)/(9!k!))>(((k+10)!)/(9!(k+1)!))×(3/5)  1>(((k+10))/((k+1)))×(3/5)  2k>25  k>((25)/2)=12.5  ⇒k=14 since we search even terms  i.e. a_(14)  is the greatest coefficient.  a_(14) =(1/5^(10) )×C_9 ^(23) ×((3/5))^(14) =((817190×3^(14) )/5^(24) )

Commented byI want to learn more last updated on 09/Jul/20

I really appreciate sir