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# (a) Simplify .(b) In the expansion of , there exist three consecutive coefficients a, b, c that satisfy a:b:c = 1:7:35. Find the

+1
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(a) Simplify $$\frac{\binom{n}{k}}{\binom{n}{k - 1}}$$.

(b) In the expansion of $$(1 + x)^n$$, there exist three consecutive coefficients a, b, c that satisfy a:b:c = 1:7:35. Find the positive integer n.

Feb 27, 2020

#1
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a.

n!

_______

(n - k)! k!

___________                          =

n!

_________________

[n - ( k - 1)]! (k - 1)!

[ (n - k) + 1 ] ! (k - 1)!

___________________  =

(n - k)!  k!

[ (n - k) + 1 ] !          ( k - 1)!

____________   *   _______  =

(n - k)!                      k!

( n - k + 1)

__________

k   Feb 27, 2020
#2
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Let  k-1  be  the  first coefficient we are looking for in the nth row of Pascal's Triangle

The second coefficient will be the  kth element  on row n

And the last coefficient  will be the (k + 1) st  element  on row  n

We  have seen  that    C(n , k)  / C( n,  k  -1)  =  (n - k + 1) / k

So...we can calculate   C( n, k + 1) / C ( n ,k)  as

n!                                         ( n - k)!  k!

________________  *             _________   =

[ n - (k+ 1)]! (k+ 1)!                          n!

( n - k)!                          k!

__________   *          ______  =

[ n - (k + 1)]!                (k + 1)!

( n - k)!                  1

________  *       _____  =

(n - k - 1)!            k + 1

n - k

________

k + 1

So   we have these two equations

(n - k + 1)  / k =   7

(n-k) / (k + 1)  =  5         simplify

n  - k + 1   =  7k

n - k  = 5 (k + 1)

n - k  + 1  = 7k

n - k  = 5k + 5

n - 8k =  -1

n - 6k  =   5      subtract these

-8k + 6k  =  - 6

-2k  = - 6

k =  3

And

n  - 8k  =  -1

n  -8(3)  =  -1

n - 24  = -1

n =  23

Check :

The coefficients  should be

C (n, k -1), C( n , k) and  C( n, k + 1)  =

C (23 , 2)  , C (23, 3)  and  C (23, 4)  =

253  ,  1771   and  8855

The ratios  are

1  :  1771/253  :   8855/253   =

1  : 7  : 35   Feb 27, 2020