find-the-greatest-coeeficient-in-the-expansion-of-3-4x-5-

Question Number 114208 by aurpeyz last updated on 17/Sep/20

f i n d t h e g r e a t e s t c o e e f i c i e n t i n t h e e x p a n s i o n

o f {(3 + 4 x)}^{- 5}

Commented by I want to learn more last updated on 17/Sep/20

Please how sir

Commented by mr W last updated on 17/Sep/20

n o m a x i m u m, n o m i n i m u m!

Commented by aurpeyz last updated on 18/Sep/20

w h y ?

Commented by mr W last updated on 18/Sep/20

{(3 + 4 x)}^{- 5}

= 3^{- 5} \times {(1 + \frac{4}{3} x)}^{- 5}

= 3^{- 5} \underset{k = 0}{\sum^{\infty}} {(- \frac{4}{3})}^{k} C_{4}^{k + 4} x^{k}

w i t h i n c r e a s i n g k, {(\frac{4}{3})}^{k} i s i n c r e a s i n g

a n d C_{4}^{k + 4} i s a s l o i n c r e a s i n g . t h e r e f o r e

t h e r e i s n o m a x / m i n f o r 3^{- 5} {(- \frac{4}{3})}^{k} C_{4}^{k + 4},

i t \to + \infty o r - \infty .

Commented by aurpeyz last updated on 18/Sep/20

i t h i n k t h e r e i s m o r e t o t h i s t o p i c t h a n i k n o w

Commented by mr W last updated on 18/Sep/20

w i t h t h a t w h a t y o u k n o w y o u c a n

s o l v e a l o t o f q u e s t i o n s, s u c h a s t h i s

o n e . j u s t a p p l y!

Commented by aurpeyz last updated on 18/Sep/20

t h a n k s a l o t S i r

Commented by aurpeyz last updated on 18/Sep/20

w h e n i t r y t o f i n d t h e g r e a t e s t c o e e f i c i e n t

u s i n g \underset{r = 0}{\sum^{\infty}} C_{r}^{5 + r - 1} {(\frac{4}{3})}^{r} {(- 1)}^{r} I arrived at a negative

value for r . I think it is a proof that there is no

greatest coefficient .

Commented by mr W last updated on 18/Sep/20

n o p r o o f n e e d e d . {i t}^{'} s c l e a r :

C_{4}^{4 + r} {(\frac{4}{3})}^{r} i s a l w a y s i n c r e a s i n g a n d

{(- 1)}^{r} = 1 f o r e v e n r a n d - 1 f o r o d d r .

s o t h e r e i s n e i t h e r m a x i m u m n o r

m i n i m u m c o e f f i c i e n t .

Commented by mr W last updated on 18/Sep/20

{i t}^{'} s s o m e t h i n g l i k e

1, - 2, 3, - 4, 5, - 6, \dots .

i t h a s n o m a x i m u m a n d n o m i n i m u m .

Commented by aurpeyz last updated on 18/Sep/20

a n y t i m e i h a v e a q u e s t i o n . h o w c a n i k n o w

w i t h o u t p r o o f t h a t t h e r e i s n o m a x o r m i n .

d i d y o u d e t e r m i n e i t b y {(\frac{4}{3})}^{r} o r t h e n a t u r e

o f t h e q u e s t i o n b e i n g {(3 + 4 x)}^{- 5} .

i h a v e n o t s e e n a n y t e x t b o o k t h a t d e a l s w i t h

g r e a t e s t c o e f f i c i e n t o f q u e s t i o n l i k e t h a t .

i t h o u g h t i t i s b e c a u s e o f t h e p o s i t i v e s i g n (+)

b e t w e e n t h e b i n o m i a l .

w h a t i f w e h a d a q u e s t i o n w i t h {(\frac{3}{4})}^{x} ?

Commented by mr W last updated on 18/Sep/20

{(- 1)}^{r} C_{4}^{4 + r} {(\frac{4}{3})}^{r} h a s n o m a x a n d n o m i n .

{(- 1)}^{r} C_{4}^{4 + r} {(\frac{3}{4})}^{r} h a s m a x a n d m i n .

Commented by aurpeyz last updated on 18/Sep/20

t h a n k y o u S i r . i w i l l a t t e m p t i t

Commented by aurpeyz last updated on 18/Sep/20

l e t s s a y i h a v e {(4 + 3 x)}^{- 5}

{(4 + 3 x)}^{- 5} = 4^{- 5} [{(- 1)}^{r} \underset{r = 0}{\sum^{\infty}} C_{r}^{4 + r} {(\frac{3}{4})}^{r}]

{(- 1)}^{r} C_{r}^{4 + r} {(\frac{3}{4})}^{r} ⩾ {(- 1)}^{r + 1} C_{r + 1}^{5 + r} (\frac{3}{4})

C_{4}^{4 + r} ⩾ (- 1) C_{4}^{5 + r} {(\frac{3}{4})}^{1}

r = \frac{- 19}{7} .

t h e n e g a t i v i t y g i v e s m e i s s u e .

i s i t t h a t a n y q u e s t i o n l i k e {(4 + 3 x)}^{- 5} [w i t h

p o s i t i v e s i g n i n b e t w e e n b i n o m i a l o f n e g a t i v e

p o w e r s] u s u a l l y d o n t h a v e m a x o r m i n ?

Commented by mr W last updated on 18/Sep/20

t o t r e a t t h e p o s i t i v e c o e f f i c i e n t s

y o u s h o u l d o n l y s e e t h e e v e n t e r m s,

i . e . r = 2 k

\dots \dots

Commented by aurpeyz last updated on 19/Sep/20

p l e a s e s h e d l i t t l e l i g h t o n t h i s . r = 2 k ? h o w

i s t h i s d o n e ?

Commented by mr W last updated on 19/Sep/20

e v e n t e r m s (r = 2 k) h a v e p o s i t i v e

c o e f f i c i e n t s, o d d t e r m s (r = 2 k + 1)

h a v e n e g a t i v e c o e f f i c i e n t s . j u s t t h i n k!

Commented by aurpeyz last updated on 20/Sep/20

o k a y

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