Answer
a)P(8) is true
--P(9) is true
--P(10) is true
b)we can form j cents postage for all j with 8 ≤ j ≤ k, where we assume
that k ≥ 10
c)Assuming the inductive hypothesis, we can
form k + 1 cents postage using just 3-cent and 5-cent stamps
d)Because k ≥ 10, we know that P(k−2) is true, that is, that
we can form k−2 cents of postage.
e)the statement is
true for every integer n greater than or equal to 8.
Work Step by Step
a) P(8) is true,
Because we can form 8 cents of postage with one 3-cent stamp and one 5-cent stamp.
--P(9) is true, because we can form 9 cents of
postage with three 3-cent stamps.
--P(10) is true, because we can form 10 cents of postage with two 5-cent stamps.
b) The statement that using just 3-cent and 5-cent stamps
we can form j cents postage for all j with 8 ≤ j ≤ k, where we assume
that k ≥ 10
c) Assuming the inductive hypothesis, we can
form k + 1 cents postage using just 3-cent and 5-cent stamps
d) Because k ≥ 10, we know that P(k−2) is true, that is, that
we can form k−2 cents of postage.
Put one more 3-cent stamp on the envelope, and we have formed k + 1 cents of postage.
e) We have completed both the basis step and the inductive
step, so by the principle of strong induction, the statement is
true for every integer n greater than or equal to 8.
