Find-the-greatest-coefficient-in-expansion-of-6-4x-3-

Question Number 178937 by Tawa11 last updated on 22/Oct/22

Find the greatest coefficient in expansion of : {(6 - 4 x)}^{- 3}

Answered by mr W last updated on 23/Oct/22

{(6 - 4 x)}^{- 3}

= 6^{- 3} {(1 - \frac{2 x}{3})}^{- 3}

= 6^{- 3} \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} {(\frac{2}{3})}^{k} x^{k}

a_{k} = 6^{- 3} {(\frac{2}{3})}^{k} C_{2}^{k + 2} = \frac{2^{k - 4} (k + 2) (k + 1)}{3^{k + 3}}

a_{k + 1} = \frac{2^{k - 3} (k + 3) (k + 2)}{3^{k + 4}}

a_{k} > a_{k + 1}

\frac{2^{k - 4} (k + 2) (k + 1)}{3^{k + 3}} > \frac{2^{k - 3} (k + 3) (k + 2)}{3^{k + 4}}

\frac{(k + 1)}{1} > \frac{2 (k + 3)}{3}

\Rightarrow k > 3

\Rightarrow a_{4} i s m a x i m u m!

a_{m a x} = a_{4} = \frac{2^{4 - 4} (4 + 2) (4 + 1)}{3^{4 + 3}} = \frac{10}{729} ✓

Commented by Tawa11 last updated on 23/Oct/22

Great, God bless you sir . I appreciare .

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