In the binomial expansion (a+x)^n where n>4, the coefficient of x^3 is twice that of x^4. Show that n=2a+3

Question

In the binomial expansion (a+x)^n where n>4, the coefficient of x^3 is twice that of x^4.

 

1) Show that n=2a+3

 

In the same expression, the coefficient of x^2 is 3/2 time the coefficient of x

 

2) Obtain another relationship between n and a. Hence find the value of n and a

 

3) State the constant term in the expansion

 

Answer 1

nC3 = 2(nC4)

n!/3!(n-3)! = 2n!/4!(n-4)!

1/3!(n-3)! = 2/4!(n-4)!

1/3! = 1/2(3!)(n-4)

1 = 1/2(n-4)

n-4 = 1/2

n = 4.5

1

11

121

1331 3rd term is 3 times 4th term

14641 3rd term is 1.5 times 4th term

15101051 3rd term is = 4th term

1615201561 3rd term < 4th term

172145452171

1,8,28, 66, 90...

there is no integer n where the 4th term has a coefficient half the 3rd term, n=4.5 would work though

for 3/2 times the 4th term, n<3

nC3 = (3/2)nC4

n!/3!(n-3)! = 3n!/2(4!)(n-4)!

1 = 3/8(n-1)

n-1= 3/8

n = 1.375