find the greatest coeeficient in the expansion (6−4x)^(−3)


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

$find the greatest coeeficient in the expansion$ ${(6 - 4 x)}^{- 3}$
	Answered by mr W last updated on 15/Sep/20

	${(6 - 4 x)}^{- 3} = 6^{- 3} {(1 - \frac{2}{3} x)}^{- 3}$ $= 6^{- 3} \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} {(\frac{2}{3})}^{k} x^{k}$ $l e t C_{2}^{k + 2} {(\frac{2}{3})}^{k} ⩾ C_{2}^{k + 3} {(\frac{2}{3})}^{k + 1}$ $\frac{(k + 2) (k + 1)}{2} ⩾ \frac{(k + 3) (k + 2)}{2} (\frac{2}{3})$ $3 (k + 1) ⩾ 2 (k + 3)$ $k ⩾ 3$ $m a x . c o e f . i s a t k = 3 :$ $6^{- 3} \times C_{2}^{5} \times {(\frac{2}{3})}^{3} = \frac{10}{729}$
	Commented by aurpeyz last updated on 16/Sep/20

	$p l s e x p l a i n l i n e 2 a n d 3 i n w o r d s . t h a n k s a l o t$ $s i r$
	Commented by mr W last updated on 16/Sep/20

	$l i n e 2 i s j u s t b i n o m i a l t h e o r e m$
	Commented by mr W last updated on 16/Sep/20

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

	$i f A_{n} i s t h e l a r g e s t, i t m e a n s :$ $A_{1} < A_{2} < . . . < A_{n - 1} < A_{n} > A_{n + 1} > A_{n + 2} > . . .$ $i . e . n i s t h e m i n i m u m o f k w h i c h$ $f u l f i l l s A_{k} ⩾ A_{k + 1}$ $o r C_{2}^{k + 2} {(\frac{2}{3})}^{k} ⩾ C_{2}^{k + 3} {(\frac{2}{3})}^{k + 1}$ $\Rightarrow k ⩾ 3$ $i t m e a n s A_{3} i s t h e l a r g e s t .$
	Commented by aurpeyz last updated on 16/Sep/20

	$i u n d e r s t a n d t h a t t h e g e n e r a l t e r m o f a$ $b i n o m i a l i s \underset{r = o}{\sum^{n}} {\overset{n}{C}}_{r} x^{n - r} y^{r} . p l s e x p l a i n t h i s \Rightarrow$ $\underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} {(\frac{2}{3})}^{k} x^{k} . h o w y o u g e t t o u s e k + 2 .$
	Commented by aurpeyz last updated on 16/Sep/20

	$s o i t h a t i w i l l b e a b l e t h i s u n d e r s t a n d i n g t o$ $s o l v e o t h e r p r o b l e m s . t h a n k s a l o t$
	Commented by mr W last updated on 16/Sep/20

	$y o u s h o u l d s t u d y t h e c a s e {(x + y)}^{n}$ $w i t h n = n e g a t i v e!$
	Commented by aurpeyz last updated on 16/Sep/20

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

	$g e n e r a l l y C_{r}^{n} = C_{n - r}^{n}$ $\Rightarrow C_{k}^{k + 2} = C_{2}^{k + 2}$
	Commented by aurpeyz last updated on 16/Sep/20

	$w o w . t h a n k s S i r . i r e a l l y a p p r e c i a t e . p e r f e c t$
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