Evaluate-the-greatest-coefficient-of-7-5x-3-

Question Number 82149 by TawaTawa last updated on 18/Feb/20

Evaluate the greatest coefficient of {(7 - 5 x)}^{- 3}

Commented by TawaTawa last updated on 18/Feb/20

Please working sir . I appreciate .

Answered by mr W last updated on 18/Feb/20

{(7 - 5 x)}^{- 3}

= 7^{- 3} {(1 - \frac{5 x}{7})}^{- 3}

= 7^{- 3} \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} {(\frac{5 x}{7})}^{k}

= \underset{k = 0}{\sum^{\infty}} \frac{5^{k}}{7^{k + 3}} C_{2}^{k + 2} x^{k}

a_{k} = \frac{5^{k}}{7^{k + 3}} C_{2}^{k + 2}

i f a_{n} i s t h e g r e a t e s t, t h e n a_{n} ⩾ a_{n + 1}, i . e .

\frac{5^{n}}{7^{n + 3}} C_{2}^{n + 2} ⩾ \frac{5^{n + 1}}{7^{n + 4}} C_{2}^{n + 3}

C_{2}^{n + 2} ⩾ \frac{5}{7} C_{2}^{n + 3}

\frac{(n + 2) (n + 1)}{2!} ⩾ \frac{5}{7} \times \frac{(n + 3) (n + 2)}{2!}

7 (n + 1) ⩾ 5 (n + 3)

n ⩾ 4

t h a t m e a n s t e r m a_{4} i s t h e g r e a t e s t .

a_{4} = \frac{5^{4}}{7^{7}} C_{2}^{6} = \frac{5^{4} \times 6 \times 5}{7^{7} \times 2!} = \frac{3 \times 5^{5}}{7^{7}} = \frac{9375}{823543}

Commented by TawaTawa last updated on 18/Feb/20

God bless you sir .

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

i s m y w o r k i n g c l e a r a n d u n d e r s t o o d ?

Commented by TawaTawa last updated on 19/Feb/20

It is only the third line am finding difficult to understand sir

Commented by TawaTawa last updated on 19/Feb/20

How to get the combination

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

y o u m u s t h a v e l e a r n t {(a + b)}^{n} . i f n o t,

t r y t o s t u d y a b o u t “ b i n o m i a l {t h e o r e m}^{″} .

w e h a v e h e r e a = 1, b = - \frac{5 x}{7} a n d n = - 3 .

Commented by TawaTawa last updated on 19/Feb/20

Yes sir . I know {(a + b)}^{n}

How can i get : \overset{k + 2}{} C_{2} {(\frac{5 x}{7})}^{k}

Thanks for your help sir

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

Commented by TawaTawa last updated on 19/Feb/20

God bless you sir . I understand now . Thanks for your time .