Answer to Question #269254 in Discrete Mathematics for Aljon

Question

1. Expand the following using the Binomial Theorem

a. (x-3y)^3

b. (2a+b^-3)^4

c. (a+b+c)²

2. Solve the recurrence relation Pn = Pn-1 + n with the initial condition p_1 = 2 using iteration.

Answer 1

a.

$$(x-3y)^3=x^3-9x^2y+27xy^2-27y^3$$

b.

$$(2a+b^{-3})^4=16a^4+32a^3b^{-3}+24a^2b^{-6}+8ab^{-9}+b^{-12}$$

c.

$$(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$$

d.

$$p_2=2+2=4$$

$$p_3=4+3=7$$

$$p_4=7+4=11$$

$$p_5=11+5=16$$

$$\vdots$$

$$(p_2-p_1)+(p_3-p_2)+(p_4-p_3)+...$$

$$+(p_{n-1}-p_{n-2})+(p_n-p_{n-1})=2+3+4+...$$

$$+(n-1)+n$$

$$p_n-p_1=\displaystyle\sum_{i=1}^ni-1$$

$$p_n-p_1=\dfrac{n(n+1)}{2}-1$$

$$p_n=1+\dfrac{n(n+1)}{2}$$