Let C(n) be the constant term in the expansion of (x + 4)n. Prove by induction that C(n) = 4n for all n is in N. (Induction on n.) The constant term of (x + 4)1 is

Question

Let C(n) be the constant term in the expansion of 

(x + 4)n.

 Prove by induction that 

C(n) = 4n

 for all n  N.

(Induction on n.) The constant term of 

(x + 4)1

 is 

 = 4.

Suppose as inductive hypothesis that the constant term of 

(x + 4)k − 1

 is 

 

   for some 

k > 1.

Then (x + 4)k = (x + 4)k − 1 ·

  

 

 

 , so its constant term is 

 

  · 4 =  

 

  , as required.

Answer 1

C(1)=4, so base case satsfies C(n)=4ⁿ when n=1.

Assume C(n)=4ⁿ. C(n+1)=(x+4)C(n)=xC(n)+4C(n)=xC(n)+4×4ⁿ. The constant in this expression is 4×4ⁿ=4ⁿ⁺¹.

But 4ⁿ⁺¹=C(n+1), which proves by induction that C(n)=4ⁿ for n∊ℕ.