find-the-greatest-coeeficient-in-the-expansion-6-4x-3-

Question Number 113796 by aurpeyz last updated on 15/Sep/20

$$\mathrm{find}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{coeeficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\: \\ $$$$\left(\mathrm{6}−\mathrm{4x}\right)^{−\mathrm{3}} \\ $$

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))

$$\left(\mathrm{6}−\mathrm{4}{x}\right)^{−\mathrm{3}} =\mathrm{6}^{−\mathrm{3}} \left(\mathrm{1}−\frac{\mathrm{2}}{\mathrm{3}}{x}\right)^{−\mathrm{3}} \\ $$$$=\mathrm{6}^{−\mathrm{3}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{k}} {x}^{{k}} \\ $$$${let}\:{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{k}} \geqslant{C}_{\mathrm{2}} ^{{k}+\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{k}+\mathrm{1}} \\ $$$$\frac{\left({k}+\mathrm{2}\right)\left({k}+\mathrm{1}\right)}{\mathrm{2}}\geqslant\frac{\left({k}+\mathrm{3}\right)\left({k}+\mathrm{2}\right)}{\mathrm{2}}\left(\frac{\mathrm{2}}{\mathrm{3}}\right) \\ $$$$\mathrm{3}\left({k}+\mathrm{1}\right)\geqslant\mathrm{2}\left({k}+\mathrm{3}\right) \\ $$$${k}\geqslant\mathrm{3} \\ $$$${max}.\:{coef}.\:{is}\:{at}\:{k}=\mathrm{3}: \\ $$$$\mathrm{6}^{−\mathrm{3}} ×{C}_{\mathrm{2}} ^{\mathrm{5}} ×\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{3}} =\frac{\mathrm{10}}{\mathrm{729}} \\ $$

Commented by aurpeyz last updated on 16/Sep/20

pls explain line 2 and 3 in words. thanks alot sir

$${pls}\:{explain}\:{line}\:\mathrm{2}\:{and}\:\mathrm{3}\:{in}\:{words}.\:{thanks}\:{alot} \\ $$$${sir} \\ $$

Commented by mr W last updated on 16/Sep/20

$${line}\:\mathrm{2}\:{is}\:{just}\:{binomial}\:{theorem} \\ $$

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.

$${if}\:{A}_{{n}} \:{is}\:{the}\:{largest},\:{it}\:{means}: \\ $$$${A}_{\mathrm{1}} <{A}_{\mathrm{2}} <…<{A}_{{n}−\mathrm{1}} <{A}_{{n}} >{A}_{{n}+\mathrm{1}} >{A}_{{n}+\mathrm{2}} >… \\ $$$${i}.{e}.\:{n}\:{is}\:{the}\:{minimum}\:{of}\:{k}\:{which} \\ $$$${fulfills}\:{A}_{{k}} \geqslant{A}_{{k}+\mathrm{1}} \\ $$$${or}\:{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{k}} \geqslant{C}_{\mathrm{2}} ^{{k}+\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{k}+\mathrm{1}} \\ $$$$\Rightarrow{k}\geqslant\mathrm{3} \\ $$$${it}\:{means}\:{A}_{\mathrm{3}} \:{is}\:{the}\:{largest}. \\ $$

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.

$${i}\:{understand}\:{that}\:{the}\:{general}\:{term}\:{of}\:{a} \\ $$$${binomial}\:{is}\:\underset{{r}={o}} {\overset{{n}} {\sum}}\overset{{n}} {{C}}_{{r}} {x}^{{n}−{r}} {y}^{{r}} .\:{pls}\:{explain}\:{this}\:\Rightarrow \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{k}} {x}^{{k}} .\:{how}\:{you}\:{get}\:{to}\:{use}\:{k}+\mathrm{2}. \\ $$

Commented by aurpeyz last updated on 16/Sep/20