91Ó°ÊÓ

Compute the first four terms in each sequence. For Exercises \(1-10,\) assume that each sequence is defined for \(n \geq 1 ;\) in Exercises \(11-14,\) assume that each sequence is defined for \(n \geq 0\) $$y_{n}=\left[(-1)^{n}\right] \sqrt{n}$$

Evaluate or simplify each expression. $$\frac{n[(n-2) !]}{(n+1) !}$$

Compute the first four terms in each sequence. For Exercises \(1-10,\) assume that each sequence is defined for \(n \geq 1 ;\) in Exercises \(11-14,\) assume that each sequence is defined for \(n \geq 0\) $$a_{n}=(n-1) /(n+1)$$

Convert each complex number to rectangular form. $$4\left(\cos \frac{5}{6} \pi+i \sin \frac{5}{6} \pi\right)$$

The 60 th term in an arithmetic sequence is \(105,\) and the common difference is \(5 .\) Find the first term.

$$\left(\begin{array}{l} 6 \\ 4 \end{array}\right)+\left(\begin{array}{l} 6 \\ 3 \end{array}\right)-\left(\begin{array}{l} 7 \\ 4 \end{array}\right)$$

Compute the first four terms in each sequence. For Exercises \(1-10,\) assume that each sequence is defined for \(n \geq 1 ;\) in Exercises \(11-14,\) assume that each sequence is defined for \(n \geq 0\) $$a_{n}=(-1)^{n} /(n+2)$$

Find the common difference in an arithmetic sequence in which \(a_{10}-a_{20}=70\).

Convert each complex number to rectangular form. $$3\left(\cos \frac{3}{2} \pi+i \sin \frac{3}{2} \pi\right)$$

Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$2^{3}+4^{3}+6^{3}+\cdots+(2 n)^{3}=2 n^{2}(n+1)^{2}$$