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 is in 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(n) = 4 n for all possible integers n in N. This statement is true when n=1 and proving that the statement is true for n=k when given that statement is true for n= k-1

Explanation:

Lets P (n) be the statement

C (n) = 4 n

if n =1

(x+4)n = (x+4)(1)=x+4

As we note that constant term is 4 C(n) = 4

4 n= 4 (1) =4

P(1) is true as C(n) = 4 n

when n=1

Let P (k-1)

C(k-1)=4(k-1)

we need to proof that p(k) is true

C(k) = C(k-1) +1)

=C(k-1)+C(1) x+4)n is linear

=4(k-1)+ C(1) P(k-1) is true

=4 k-4 +4 f(1)=4

=4 k

So p(k) is true

By the principle of mathematical induction, p(n) is true for all positive integers n