Chapter 9

In Exercises \(1-9\), simplify the given expression. $$ (3 !)^{2} $$

In Exercises \(1-13,\) write out the first four terms of the given sequence. \(a_{n}=2^{n}-1, n \geq 0\)

Prove each assertion using the Principle of Mathematical Induction. $$ \sum_{j=1}^{n} j^{2}=\frac{n(n+1)(2 n+1)}{6} $$

In Exercises \(1-9\), simplify the given expression. $$ \frac{10 !}{7 !} $$

Write out the first four terms of the given sequence. \(d_{j}=(-1)^{\frac{j(j+1)}{2}}, j \geq 1\)

Prove each assertion using the Principle of Mathematical Induction. $$ \sum_{j=1}^{n} j^{3}=\frac{n^{2}(n+1)^{2}}{4} $$

In Exercises \(1-9\), simplify the given expression. $$ \frac{7 !}{2^{3} 3 !} $$

Write out the first four terms of the given sequence. \(\\{5 k-2\\}_{k=1}^{\infty}\)

Prove each assertion using the Principle of Mathematical Induction. \(2^{n}>500 n\) for \(n>12\)

In Exercises \(1-9\), simplify the given expression. $$ \frac{9 !}{4 ! 3 ! 2 !} $$

Write out the first four terms of the given sequence. \(\left\\{\frac{n^{2}+1}{n+1}\right\\}_{n=0}^{\infty}\)

In Exercises \(1-9\), simplify the given expression. $$ \frac{(n+1) !}{n !}, n \geq 0 $$

In Exercises \(1-9\), simplify the given expression. $$ \frac{(k-1) !}{(k+2) !}, k \geq 1 $$

Write out the first four terms of the given sequence. \(\left\\{\frac{\ln (n)}{n}\right\\}_{n=1}^{\infty}\)

Use the Product Rule for Logarithms to show \(\log \left(x^{n}\right)=n \log (x)\) for all real numbers \(x>0\) and all natural numbers \(n \geq 1\).

In Exercises \(1-9\), simplify the given expression. $$ \left(\begin{array}{l} 8 \\ 3 \end{array}\right) $$

Write out the first four terms of the given sequence. \(a_{1}=3, a_{n+1}=a_{n}-1, n \geq 1\)

Prove each assertion using the Principle of Mathematical Induction. \(\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]^{n}=\left[\begin{array}{cc}a^{n} & 0 \\ 0 & b^{n}\end{array}\right]\) for \(n \geq 1\)

Write out the first four terms of the given sequence. \(d_{0}=12, d_{m}=\frac{d_{m-1}}{100}, m \geq 1\)

In Exercises \(1-9\), simplify the given expression. $$ \left(\begin{array}{c} 117 \\ 0 \end{array}\right) $$

Prove Equations 9.1 and 9.2 for the case of geometric sequences. That is: (a) For the sequence \(a_{1}=a, a_{n+1}=r a_{n}, n \geq 1,\) prove \(a_{n}=a r^{n-1}, n \geq 1\) (b) \(\sum_{j=1}^{n} a r^{n-1}=a\left(\frac{1-r^{n}}{1-r}\right),\) if \(r \neq 1, \sum_{j=1}^{n} a r^{n-1}=n a,\) if \(r=1\).

In Exercises \(1-9\), simplify the given expression. $$ \left(\begin{array}{c} n \\ n-2 \end{array}\right), n \geq 2 $$

Write out the first four terms of the given sequence. \(b_{1}=2, b_{k+1}=3 b_{k}+1, k \geq 1\)

In Exercises \(9-16\), rewrite the sum using summation notation. $$ 8+11+14+17+20 $$

Prove that the determinant of a lower triangular matrix is the product of the entries on the main diagonal. (See Exercise 8.3 .1 in Section 8.3.) Use this result to then show \(\operatorname{det}\left(I_{n}\right)=1\) where \(I_{n}\) is the \(n \times n\) identity matrix.

In Exercises \(10-13,\) use Pascal's Triangle to expand the given binomial. $$ (x+2)^{5} $$

Write out the first four terms of the given sequence. \(c_{0}=-2, c_{j}=\frac{c_{j-1}}{(j+1)(j+2)}, j \geq 1\)

Rewrite the sum using summation notation. $$ 1-2+3-4+5-6+7-8 $$

In Exercises \(10-13,\) use Pascal's Triangle to expand the given binomial. $$ (2 x-1)^{4} $$

Write out the first four terms of the given sequence. \(a_{1}=117, a_{n+1}=\frac{1}{a_{n}}, n \geq 1\)

Rewrite the sum using summation notation. $$ x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7} $$

In Exercises \(10-13,\) use Pascal's Triangle to expand the given binomial. $$ \left(\frac{1}{3} x+y^{2}\right)^{3} $$

Rewrite the sum using summation notation. $$ 1+2+4+\cdots+2^{29} $$

In Exercises \(10-13,\) use Pascal's Triangle to expand the given binomial. $$ \left(x-x^{-1}\right)^{4} $$

Write out the first four terms of the given sequence. \(F_{0}=1, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}, n \geq 2\) (This is the famous Fibonacci Sequence )

Rewrite the sum using summation notation. $$ 2+\frac{3}{2}+\frac{4}{3}+\frac{5}{4}+\frac{6}{5} $$

In Exercises \(14-21\) determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference \(d ;\) if it is geometric, find the common ratio \(r\). \(\\{3 n-5\\}_{n=1}^{\infty}\)

In Exercises 14 - 17 , use Pascal's Triangle to simplify the given power of a complex number. $$ (1+2 i)^{4} $$

Rewrite the sum using summation notation. $$ -\ln (3)+\ln (4)-\ln (5)+\cdots+\ln (20) $$

Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference \(d ;\) if it is geometric, find the common ratio \(r\). \(a_{n}=n^{2}+3 n+2, n \geq 1\)

In Exercises 14 - 17 , use Pascal's Triangle to simplify the given power of a complex number. $$ (-1+i \sqrt{3})^{3} $$

Rewrite the sum using summation notation. $$ 1-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+\frac{1}{25}-\frac{1}{36} $$

Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference \(d ;\) if it is geometric, find the common ratio \(r\). \(\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, \ldots\)

In Exercises 14 - 17 , use Pascal's Triangle to simplify the given power of a complex number. $$ \left(\frac{\sqrt{3}}{2}+\frac{1}{2} i\right)^{3} $$

Rewrite the sum using summation notation. $$ \frac{1}{2}(x-5)+\frac{1}{4}(x-5)^{2}+\frac{1}{6}(x-5)^{3}+\frac{1}{8}(x-5)^{4} $$

Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference \(d ;\) if it is geometric, find the common ratio \(r\). \(\left\\{3\left(\frac{1}{5}\right)^{n-1}\right\\}_{n=1}^{\infty}\)

In Exercises 14 - 17 , use Pascal's Triangle to simplify the given power of a complex number. $$ \left(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i\right)^{4} $$

In Exercises 17 - 28 , use the formulas in Equation 9.2 to find the sum. $$ \sum_{n=1}^{10} 5 n+3 $$

Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference \(d ;\) if it is geometric, find the common ratio \(r\). \(17,5,-7,-19, \ldots\)

In Exercises \(18-22,\) use the Binomial Theorem to find the indicated term. The term containing \(x^{3}\) in the expansion \((2 x-y)^{5}\)