91Ó°ÊÓ

Determine whether each sequence is arithmetic. If it is, find the common difference. $$9,6,3,0,-3,-6, \dots$$

Prove statement using mathematical induction for all positive integers \(n.\) \(n^{3}-n\) is divisible by 3

Determine whether each sequence is arithmetic. If it is, find the common difference. $$120,60,30,15, \ldots$$

Find the first four terms of each sequence described. Determine whether the sequence is arithmetic, and if so, find the common difference. $$a_{n}=-4 n+5$$

Find the general, or \(n\)th, term of each arithmetic sequence given the first term and the common difference. $$a_{1}=11 \quad d=5$$

Find the general, or \(n\)th, term of each arithmetic sequence given the first term and the common difference. $$a_{1}=5 \quad d=11$$

Write an expression for the \(n\) th term of the given sequence. Assume \(n\) starts at 1. $$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots$$

Geometry. Prove, with mathematical induction, that the sum of the measures of the interior angles in degrees of a regular polygon of \(n\) sides is given by the formula \((n-2)\left(180^{\circ}\right)\) for \(n \geq 3\) (Hint: Divide a polygon into triangles. For example, a four-sided polygon can be divided into two triangles. A five-sided polygon can be divided into three triangles. A six-sided polygon can be divided into four triangles, and so on.)

Apply mathematical induction to prove $$\sum_{k=1}^{n} k^{4}=\frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)}{30}$$

Simplify each ratio of factorials. $$\frac{97 !}{93 !}$$