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.

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

Step-by-step 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