Use mathematical induction to prove that the statement is true for the positive integer $n$.

$1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=$ $\frac{n}{6}(n+1)(2 n+7)$

Write the complete binomial expansion for the given powers of a binomial.

$$(x-5)^3$$

Determine whether the equation is true or false.

$$\sum_{x=1}^8(2 x)=\sum_{y=3}^9(2 y-4)$$

Use mathematical induction to prove that the statement is true for the positive integer $n$.

$$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

Use the addition rule to solve the problem.

If $P(C)=0.3, P(D)=0.5$, and $P(C \cap D)=0.2$, then what is $P(C \cup D)$ ?

Evaluate the expression.

$$P(44,0)$$

Identify the sequence as arithmetic, geometric, or neither.

$$5,1, \frac{1}{5}, \frac{1}{25}, \ldots$$

Find the first five terms of the infinite sequence whose $nth$ term is given.

$$c_n=\frac{(-2)^{2 n-1}}{(n-1) !}$$

Solve the problem using the idea of labeling.

How many distinct triangles are determined by six points lying on a circle, where the vertices of each triangle are chosen from the six points?

Determine whether the equation is true or false.

$$\sum_{x=1}^5(2 x-1)=\sum_{y=3}^7(2 y-1)$$

Evaluate the expression.

$$P(99,0)$$

Solve the problem using the idea of labeling.

How many distinct chords (line segments with endpoints on the circle) are determined by three points lying on a circle? By four points? By five points? By $n$ points?

Find the first five terms of the infinite sequence whose $nth$ term is given.

$$c_n=\frac{(-1)^n}{(n+1) !}$$

Identify the sequence as arithmetic, geometric, or neither.

$$1,4,9,16,25, \ldots$$