Find-th-greatest-coefficients-in-the-expansion-of-3a-5b-18-2-If-three-consecutive-coefficient-of-1-x-n-are-28-56-70-find-the-value-of-n-

Question Number 75242 by peter frank last updated on 08/Dec/19

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

i n t h e e x p a n s i o n o f

{(3 a + 5 b)}^{18}

2) I f t h r e e c o n s e c u t i v e

c o e f f i c i e n t o f {(1 + x)}^{n} a r e 28, 56, 70 .

f i n d t h e v a l u e o f n

Answered by mr W last updated on 08/Dec/19

(1)

{(3 a + 5 b)}^{18} = \underset{k = 0}{\sum^{18}} C_{k}^{18} {(3 a)}^{k} {(5 b)}^{18 - k}

= \underset{k = 0}{\sum^{18}} 3^{k} 5^{18 - k} C_{k}^{18} a^{k} b^{18 - k}

a_{k} = 3^{k} 5^{18 - k} C_{k}^{18}

a_{k + 1} = 3^{k + 1} 5^{17 - k} C_{k + 1}^{18}

\frac{a_{k + 1}}{a_{k}} = \frac{3^{k + 1} 5^{17 - k} C_{k + 1}^{18}}{3^{k} 5^{18 - k} C_{k}^{18}} = \frac{3 (18 - k)}{5 (k + 1)} < 1

5 k + 5 > 54 - 3 k

k > \frac{49}{8} = 6.12 = 7 \Rightarrow a_{8} < a_{7}

m a x . a_{k} i s a_{7} = 3^{7} 5^{11} C_{7}^{18} = 3398392968750000

Commented by peter frank last updated on 09/Dec/19

t h a n k y o u v e r y m u c h

Answered by mr W last updated on 08/Dec/19

(2)

C_{k}^{n} = \frac{n!}{k! (n - k)!} = 28

C_{k + 1}^{n} = \frac{n!}{(k + 1)! (n - k - 1)!} = 56

\frac{n - k}{k + 1} = \frac{56}{28} = 2

\Rightarrow n = 3 k + 2

C_{k + 2}^{n} = \frac{n!}{(k + 2)! (n - k - 2)!} = 70

\frac{n - k - 1}{k + 2} = \frac{70}{56}

\Rightarrow 28 n = 63 k + 98

\Rightarrow 28 (3 k + 2) = 63 k + 98

\Rightarrow k = 2

\Rightarrow n = 8

Commented by peter frank last updated on 09/Dec/19

t h a n k y o u v e r y m u c h