91Ó°ÊÓ

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ -2-3-4-\cdots-(n+1)=-\frac{1}{2} n(n+3) $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{f_{n}\right\\}=\left\\{3^{2 n}\right\\} $$

List the first five terms of each sequence. \(\left\\{a_{n}\right\\}=\left\\{\frac{n}{n+2}\right\\}\)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{1}{3} n(n+1)(n+2) $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{t_{n}\right\\}=\left\\{\frac{3^{n-1}}{2^{n}}\right\\} $$

Expand each expression using the Binomial Theorem. $$ (x+1)^{5} $$

In Problems 17-24, find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=2 ; \quad d=3 $$

Expand each expression using the Binomial Theorem. $$ (x-1)^{5} $$

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=-2 ; \quad d=4 $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{u_{n}\right\\}=\left\\{\frac{2^{n}}{3^{n-1}}\right\\} $$