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