Solution for Section 4.3.3

Solution for Section 4.3 Question 3

3. Let P(n) be the claim that b_n = 7 - 4n for all integers n0.

P(0) is the statement: b₀ = 7 - 4·0.

P(k) is the statement: b_k = 7 - 4k.

P(k+1) is the statement: b_k+1 = 7 - 4(k+1).

For a proof by induction, you first need to check that the statement P(0) is true. Then assume that P(k) is true and use this to show that P(k + 1) is true.

The statement P(0) is true since we are told in the question that b₀ = 7 = 7 - 4·0.

Now assume that the statement P(k) is true. We now need to show that the left-hand side of P(k+1) is equal to the right-hand side of P(k+1). We are told in the question that b_i = b_i-1 - 4 for all integers i1, so we can use that fact.

L.H.S. of P(k+1)	=	b_k+1
	=	b_k - 4
	=	7 - 4k - 4 (since we are assuming that P(k) is true)
	=	7 - 4(k + 1)
	=	R.H.S. of P(k+1)

Therefore, b_n = 7 - 4n is a general formula for this sequence for all integers n0.

Back to Section 4.3