Answer to Question #348708 in Discrete Mathematics for Heillie

Question

Find the value of a2 for the recurrence relation an=17an-1+30n, where a0=3

Find the value of a3 for the recurrence relation an=17an-1+30n, where a0=3

Find the value of a1 for the recurrence relation an=17an-1+30n, where a0=3

What would be the hypothesis of the mathematical induction for x(x + 1) < x! , where x ≥ 7?

If P(k) = k2(k + 2)(k – 1) is true, then what is P (k + 1)?

Answer 1

1.

$$a_1=17(3)+30(1)=81$$

$$a_2=17(81)+30(2)=1437$$

2.

$$a_3=17(1437)+30(3)=24519$$

3.

$$a_1=17(3)+30(1)=81$$

4.

It is assumed that at $x = k,$ $P(k)$ holds

$$k(k+1)<k!$$

5.

If $P(k) = k^2(k + 2)(k – 1)$ is true, then it must be shown that $P(k + 1)$ is true, namely, that

$$(k+1)^2(k+1 + 2)(k+1 – 1)$$

is also true.

Simplify

$$k(k+1)^2(k+3)$$

is also true.