Chapter 10

Evaluate each binomial coefficient. $$\left(\begin{array}{l} 7 \\ 3 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) $$n^{2} \leq n^{3}$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$1,3,9,27, \ldots$$

Determine whether each sequence is arithmetic. If it is, find the common difference. $$2,5,8,11,14, \dots$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=n$$

Evaluate each binomial coefficient. $$\left(\begin{array}{l} 8 \\ 2 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) $$\text { If } 0< x<1, \text { then } 0< x^{n}<1$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$2,4,8,16, \dots$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=n^{2}$$

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

Evaluate each binomial coefficient. $$\left(\begin{array}{c} 10 \\ 8 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) $$2 n \leq 2^{n}$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=2 n-1$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$1,4,9,16,25, \dots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{c} 23 \\ 21 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) $$5^{n}<5^{n+1}$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \dots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{l} 17 \\ 0 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) $$\left.n !>2^{n} \quad n \geq 4 \quad \text { (Show it is true for } n=4, \text { instead of } n=1 .\right)$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{n}{(n+1)}$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$8,4,2,1, \dots$$

Determine whether each sequence is arithmetic. If it is, find the common difference. $$3.33,3.30,3.27,3.24, \ldots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{c} 100 \\ 0 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) $$(1+c)^{n} \geq n c \quad c>1$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{(n+1)}{n}$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$8,-4,2,-1, \dots$$

Determine whether each sequence is arithmetic. If it is, find the common difference. $$0.7,1.2,1.7,2.2, \ldots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{l} 99 \\ 99 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) \(n(n+1)(n-1)\) is divisible by 3

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{2^{n}}{n !}$$

Determine whether each sequence is geometric. If it is, find the common ratio. $$800,1360,2312,3930.4, \ldots$$

Determine whether each sequence is arithmetic. If it is, find the common difference. $$4, \frac{14}{3}, \frac{16}{3}, 6, \dots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{c} 52 \\ 52 \end{array}\right)$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{n !}{(n+1) !}$$

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

Determine whether each sequence is geometric. If it is, find the common ratio. $$7,15.4,33.88,74.536, \ldots$$

Determine whether each sequence is arithmetic. If it is, find the common difference. $$2, \frac{7}{3}, \frac{8}{3}, 3, \ldots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{c} 48 \\ 45 \end{array}\right)$$

Prove statement using mathematical induction for all positive integers \(n.\) \(n^{2}+3 n\) is divisible by 2

Write the first five terms of each geometric series. $$a_{1}=6 \quad r=3$$

Determine whether each sequence is arithmetic. If it is, find the common difference. $$10^{1}, 10^{2}, 10^{3}, 10^{4}, \ldots$$

Evaluate each binomial coefficient. $$\left(\begin{array}{l} 29 \\ 26 \end{array}\right)$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=(-1)^{n+1} n^{2}$$

Prove statement using mathematical induction for all positive integers \(n.\) \(n(n+1)(n+2)\) is divisible by 6

Write the first five terms of each geometric series. $$a_{1}=17 \quad r=2$$

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

Expand each expression using the Binomial theorem. $$(x+2)^{4}$$

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{(-1)^{n}}{(n+1)(n+2)}$$

Prove statement using mathematical induction for all positive integers \(n.\) $$2+4+6+8+\dots+2 n=n(n+1)$$

Write the first five terms of each geometric series. $$a_{1}=1 \quad r=-4$$