Section 1
Sequences and Series
Fill in the blank(s).The function values $a_{1}, a_{2}, a_{3}, a_{4}, \ldots, a_{n}, \ldots$ are called the _____ of a sequence.
Fill in the blank(s).If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, then the sequence is defined _____.
Fill in the blank(s).For the sum $\sum_{i=1}^{n} a_{i}, i$ is called the _____ of summation, $n$ is the _____ of summation, and 1 is the _____ of summation.
Fill in the blank(s).The sum of the terms of a finite or an infinite sequence is called a _____.
Which describes an infinite sequence? a finite sequence?(a) The domain consists of the first $n$ positive integers.(b) The domain consists of the set of positive integers.
Write $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6$ in factorial notation.
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=2 n-5$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=4 n-7$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=3^{n}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\left(\frac{1}{2}\right)^{n}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\left(-\frac{1}{2}\right)^{n}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=(-2)^{n}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{n+1}{n}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{n}{n+1}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{n}{n^{2}+1}$$
Write the first five terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{2 n}{n+1}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=\frac{1+(-1)^{n}}{n}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=\frac{1+(-1)^{n}}{2 n}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=1-\frac{1}{2^{n}}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=\frac{3^{n}}{4^{n}}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=\frac{1}{n^{3 / 2}}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=\frac{1}{\sqrt{n}}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=\frac{(-1)^{n}}{n^{2}}$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=(2 n-1)(2 n+1)$$
Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume $n$ begins with 1.) $$a_{n}=n(n-1)(n-2)$$
Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume $n$ begins with 1.)$$a_{n}=2(3 n-1)+5$$
Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume $n$ begins with 1.)$$a_{n}=2 n(n+1)-7$$
Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume $n$ begins with 1.)$$a_{n}=1+\frac{n+1}{n}$$
Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume $n$ begins with 1.)$$a_{n}=\frac{4 n^{2}}{n+2}$$
Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume $n$ begins with 1.)$$a_{n}=(-1)^{n}+1$$
Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume $n$ begins with 1.)$$a_{n}=(-1)^{n+1}+8$$
Find the indicated term of the sequence.$$\begin{aligned}&a_{n}=\frac{n^{2}}{n^{2}+1}\\&a_{10}=\end{aligned}$$
Find the indicated term of the sequence.$$\begin{aligned}&a_{n}=\frac{n^{2}}{2 n+1}\\&a_{5}=\end{aligned}$$
Find the indicated term of the sequence.$$\begin{aligned}&a_{n}=(-1)^{n}(3 n-2)\\&a_{25}=\end{aligned}$$
Find the indicated term of the sequence.$$\begin{aligned}&a_{n}=(-1)^{n-1}[n(n-1)]\\&a_{16}=\end{aligned}$$
Find the indicated term of the sequence.$$\begin{aligned}&a_{n}=\frac{2^{n+1}}{2^{n}+1}\\&a_{7}=\end{aligned}$$
Find the indicated term of the sequence.$$\begin{aligned}&a_{n}=\frac{3^{n}}{3^{n}+1}\\&a_{6}=\end{aligned}$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$3,8,13,18,23, \dots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$3,7,11,15,19, \ldots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$7,13,19,25,31, \ldots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$9,11,13,15,17, \ldots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$0,3,8,15,24, \ldots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$4,7,12,19,28, \dots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$\frac{2}{1}, \frac{3}{3}, \frac{4}{5}, \frac{5}{7}, \frac{6}{9}, \dots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$\frac{1}{2}, \frac{-1}{4}, \frac{1}{8}, \frac{-1}{16}, \dots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$\frac{1}{3},-\frac{2}{9}, \frac{4}{27},-\frac{8}{81}, \dots$$
$$1+\frac{1}{1}, 1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \dots$$$$1+\frac{1}{1}, 1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \dots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$1+\frac{1}{3}, 1+\frac{1}{6}, 1+\frac{1}{11}, 1+\frac{1}{18}, 1+\frac{1}{27}, \ldots$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$1,3,1,3,1, . . .$$
Write an expression for the apparent $n$ th term of the sequence. (Assume $n$ begins with $1 .$)$$1,-1,1,-1,1, \ldots$$
Write the first five terms of the sequence defined recursively.$$a_{1}=28, a_{k}=a_{k-1}-4$$
Write the first five terms of the sequence defined recursively.$$a_{1}=15, a_{k}=a_{k-1}+3$$
Write the first five terms of the sequence defined recursively.$$a_{1}=3, a_{k+1}=2\left(a_{k}-1\right)$$
Write the first five terms of the sequence defined recursively.$$a_{1}=32, a_{k+1}=\frac{1}{2} a_{k}$$
Write the first five terms of the sequence defined recursively.$$a_{0}=1, a_{1}=3, a_{k}=a_{k-2}+a_{k-1}$$
Write the first five terms of the sequence defined recursively.$$a_{0}=-1, a_{1}=5, a_{k}=a_{k-2}+a_{k-1}$$
Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$ th term of the sequence as a function of $n .$ (Assume $n$ begins with 1.)$$a_{1}=6, a_{k+1}=a_{k}+2$$
Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$ th term of the sequence as a function of $n .$ (Assume $n$ begins with 1.)$$a_{1}=25, a_{k+1}=a_{k}-5$$
Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$ th term of the sequence as a function of $n .$ (Assume $n$ begins with 1.)$$a_{1}=81, a_{k+1}=\frac{1}{3} a_{k}$$
Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$ th term of the sequence as a function of $n .$ (Assume $n$ begins with 1.)$$a_{1}=14, a_{k+1}=-2 a_{k}$$
Write the first five terms of the sequence(a) using the table feature of a graphing utility and(b) algebraically. (Assume $n$ begins with 0.)$$a_{n}=\frac{1}{n !}$$
Write the first five terms of the sequence(a) using the table feature of a graphing utility and(b) algebraically. (Assume $n$ begins with 0.)$$a_{n}=\frac{1}{(n+1) !}$$
Write the first five terms of the sequence(a) using the table feature of a graphing utility and(b) algebraically. (Assume $n$ begins with 0.)$$a_{n}=\frac{n^{2}}{(n+1) !}$$
Write the first five terms of the sequence(a) using the table feature of a graphing utility and(b) algebraically. (Assume $n$ begins with 0.)$$a_{n}=\frac{n^{3}}{(n+2) !}$$
Write the first five terms of the sequence(a) using the table feature of a graphing utility and(b) algebraically. (Assume $n$ begins with 0.)$$a_{n}=\frac{(-1)^{2 n}}{(2 n) !}$$
Write the first five terms of the sequence(a) using the table feature of a graphing utility and(b) algebraically. (Assume $n$ begins with 0.)$$a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}$$
Simplify the factorial expression.$$\frac{2 !}{4 !}$$
Simplify the factorial expression.$$\frac{5 !}{7 !}$$
Simplify the factorial expression.$$\frac{12 !}{4 ! \cdot 8 !}$$
Simplify the factorial expression.$$\frac{10 !}{5 ! \cdot 3 !}$$
Simplify the factorial expression.$$\frac{(n+3) !}{n !}$$
Simplify the factorial expression.$$\frac{(n+2) !}{n !}$$
Simplify the factorial expression.$$\frac{(2 n-1) !}{(2 n+1) !}$$
Simplify the factorial expression.$$\frac{(2 n-2) !}{(2 n) !}$$
Match the sequence with its graph. [The graphs are Iabeled (a), (b), (c), and (d).]$$a_{n}=\frac{8}{n+1}$$
Match the sequence with its graph. [The graphs are Iabeled (a), (b), (c), and (d).]$$a_{n}=\frac{8 n}{n+1}$$
Match the sequence with its graph. [The graphs are Iabeled (a), (b), (c), and (d).]$$a_{n}=\frac{2^{n+2}}{2 n !}$$
Match the sequence with its graph. [The graphs are Iabeled (a), (b), (c), and (d).]$$a_{n}=\frac{4^{n}}{n !}$$
Use a graphing utility to graph the first 10 terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{2}{3} n$$
Use a graphing utility to graph the first 10 terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{1}{2} n+3$$
Use a graphing utility to graph the first 10 terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=16(-0.5)^{n-1}$$
Use a graphing utility to graph the first 10 terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=8(-0.75)^{n-1}$$
Use a graphing utility to graph the first 10 terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{2 n}{n+1}$$
Use a graphing utility to graph the first 10 terms of the sequence. (Assume $n$ begins with 1.)$$a_{n}=\frac{3 n^{2}}{n^{2}+1}$$
Find the sum.$$\sum_{i=1}^{5}(2 i+1)$$
Find the sum.$$\sum_{i=1}^{6}(3 i-1)$$
Find the sum.$$\sum_{i=0}^{6} 4 i^{2}$$
Find the sum.$$\sum_{i=0}^{5} 3 i^{2}$$
Find the sum.$$\sum_{j=3}^{5} \frac{1}{j^{2}-3}$$
Find the sum.$$\sum_{j=3}^{5} \frac{1}{j+1}$$
Find the sum.$$\sum_{k=1}^{4} 10$$
Find the sum.$$\sum_{k=1}^{5} 4$$
Find the sum.$$\sum_{i=2}^{5}\left[(i-1)^{3}+(i+1)^{2}\right]$$
Find the sum.$$\sum_{k=2}^{7}\left[(k+1)+(k-3)^{2}\right]$$
Find the sum.$$\sum_{i=0}^{4} 2^{i}$$
Find the sum.$$\sum_{j=0}^{4}(-2)^{j}$$
Use a graphing utility to find the sum.$$\sum_{j=1}^{6}(24-3 j)$$
Use a graphing utility to find the sum.$$\sum_{j=1}^{10} \frac{6}{3 j+1}$$
Use a graphing utility to find the sum.$$\sum_{k=0}^{4} \frac{(-1)^{k}}{(k+1) !}$$
Use a graphing utility to find the sum.$$\sum_{k=0}^{4} \frac{(-1)^{k}}{k !}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(9)}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\cdots+\frac{5}{1+15}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\left[2\left(\frac{1}{8}\right)+3\right]+\left[2\left(\frac{2}{8}\right)+3\right]+\cdots+\left[2\left(\frac{8}{8}\right)+3\right]$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\left[1-\left(\frac{1}{6}\right)^{2}\right]+\left[1-\left(\frac{2}{6}\right)^{2}\right]+\cdots+\left[1-\left(\frac{6}{6}\right)^{2}\right]$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$-3+9-27+81-243+729$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdot \cdot \cdot-\frac{1}{128}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdots \cdot-\frac{1}{20^{2}}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\frac{1}{1 \cdot 3}-\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}-\frac{1}{4 \cdot 6}+\cdot \cdot \cdot-\frac{1}{10 \cdot 12}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$$
Use sigma notation to write the sum. Then use a graphing utility to find the sum.$$\frac{1}{2}+\frac{2}{4}+\frac{6}{8}+\frac{24}{16}+\frac{120}{32}+\frac{720}{64}$$
Find the indicated partial sum of the series.$\sum_{i=1}^{\infty} \frac{7}{5^{i}}$Fourth partial sum
Find the indicated partial sum of the series.$\sum_{i=1}^{\infty} \frac{2}{3^{i}}$Fifth partial sum
Find the indicated partial sum of the series.$\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n}$Third partial sum
Find the indicated partial sum of the series.$\sum_{n=1}^{\infty} 8\left(-\frac{1}{4}\right)^{n}$Fourth partial sum
Find (a) the fourth partial sum and (b) the sum of the infinite series.$$\sum_{i=1}^{\infty} \frac{6}{10^{i}}$$
Find (a) the fourth partial sum and (b) the sum of the infinite series.$$\sum_{k=1}^{\infty} \frac{4}{10^{k}}$$
Find (a) the fourth partial sum and (b) the sum of the infinite series.$$\sum_{k=1}^{\infty} 7\left(\frac{1}{10}\right)^{k}$$
Find (a) the fourth partial sum and (b) the sum of the infinite series.$$\sum_{i=1}^{\infty} 2\left(\frac{1}{10}\right)^{i}$$
A deposit of $5000$dollar is made in an account that earns $3 \%$ interest compounded quarterly. The balance in the account after $n$ quarters is given by $$A_{n}=5000\left(1+\frac{0.03}{4}\right)^{n}, \quad n=1,2,3, \ldots$$(a) Compute the first eight terms of this sequence.(b) Find the balance in this account after 10 years by computing the $40$th term of the sequence.
An investor deposits $10,000$dollar in an account that earns $3.5 \%$ interest compounded monthly. The balance in the account after $n$ months is given by $$A_{n}=10,000\left(1+\frac{0.035}{12}\right)^{n}, \quad n=1,2,3, . . .$$(a) Write the first eight terms of the sequence.(b) Find the balance in the account after 5 years by computing the 60 th term of the sequence.(c) Is the balance after 10 years twice the balance after 5 years? Explain.
A landlocked lake has been selected to be stocked in the year 2015 with 5500 trout, and to be restocked each year thereafter with 500 trout. Each year the fish population declines $25 \%$ due to harvesting and other natural causes.(a) Write a recursive sequence that gives the population $p_{n}$ of trout in the lake in terms of the year $n,$ with $n=0$ corresponding to 2015.(b) Use the recursion formula from part (a) to find the numbers of trout in the lake for $n=1,2,3$ and $4 .$ Interpret these values in the context of the situation.(c) Use a graphing utility to find the number of trout in the lake as time passes infinitely. Explain your result.
The revenues $R_{n}$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table. (Source: Netflix, Inc.)(a) Use a graphing utility to graph the data. Let $n$ represent the year, with $n=8$ corresponding to 2008.(b) Use the regression feature of the graphing utility to find a linear sequence and a quadratic sequence that model the data. Identify the coefficient of determination for each model.(c) Graph each model with the data. Decide which model is a better fit for the data. Explain.(d) Use the model you chose in part (c) to predict the revenue of Netflix in $2017 .$ Does your answer seem reasonable? Explain.(e) Use your model from part (c) to find when the revenue will reach 15 billion dollars.(f) Use the model you chose in part (c) to approximate the total revenue from 2008 through 2013 . Compare this sum with the result of adding the revenues shown in the table.
Determine whether the statement is true or false. Justify your answer.$$\sum_{i=1}^{4}\left(i^{2}+2 i\right)=\sum_{i=1}^{4} i^{2}+2 \sum_{i=1}^{4} i$$
Determine whether the statement is true or false. Justify your answer.$$\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}$$
Use the Fibonacci sequence. (See Example 5.)Write the first 12 terms of the Fibonacci sequence $a_{n}$ and the first 10 terms of the sequence given by$$b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n>0$$.
Use the Fibonacci sequence. (See Example 5.)Using the definition of $b_{n}$ given in Exercise $127,$ show that $b_{n}$ can be defined recursively by$$b_{n}=1+\frac{1}{b_{n-1}}$$.
Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with $n$ th term $a_{n}$.Use the table feature of a graphing utility to find the first five terms of the sequence.
Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with $n$ th term $a_{n}$.Do you recognize the terms of the sequence in Exercise $129 ?$ What sequence is it?
Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with $n$ th term $a_{n}$.Find expressions for $a_{n+1}$ and $a_{n+2}$ in terms of $n$.
Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with $n$ th term $a_{n}$.Use the result from Exercise 131 to show that $a_{n+2}=a_{n+1}+a_{n} .$ Is this result the same as your answer to Exercise $129 ?$ Explain.
Write the first five terms of the sequence.$$a_{n}=\frac{x^{n}}{n !}$$
Write the first five terms of the sequence.$$a_{n}=\frac{x^{2}}{n^{2}}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n} x^{2 n+1}}{2 n+1}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n} x^{n+1}}{n+1}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n} x^{2 n}}{(2 n) !}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n} x^{2 n+1}}{(2 n+1) !}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n} x^{n}}{n !}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n} x^{n+1}}{(n+1) !}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n+1}(x+1)^{n}}{n !}$$
Write the first five terms of the sequence.$$a_{n}=\frac{(-1)^{n}(x-1)^{n}}{(n+1) !}$$
Write the first five terms of the sequence. Then find an expression for the $n$ th partial sum.$$a_{n}=\frac{1}{2 n}-\frac{1}{2 n+2}$$
Write the first five terms of the sequence. Then find an expression for the $n$ th partial sum.$$a_{n}=\frac{1}{n}-\frac{1}{n+1}$$
Write the first five terms of the sequence. Then find an expression for the $n$ th partial sum.$$a_{n}=\frac{1}{n+1}-\frac{1}{n+2}$$
Write the first five terms of the sequence. Then find an expression for the $n$ th partial sum.$$a_{n}=\frac{1}{n}-\frac{1}{n+2}$$
Does every finite series whose terms are integers have a finite sum? Explain.
The graph represents the first six terms of a sequence.(a) Write the first six terms of the sequence.(b) Write an expression for the apparent $n$ th term $a_{n}$ of the sequence.(c) Use sigma notation to represent the partial sum of the first 50 terms of the sequence.
Find, if possible, (a) $A-B,$ (b) $2 B-3 A,$ (c) $A B,$ and (d) $B A.$$$A=\left[\begin{array}{l}6 \\3\end{array}\right], B=\left[\begin{array}{r}4 \\-3\end{array}\right]$$
Find, if possible, (a) $A-B,$ (b) $2 B-3 A,$ (c) $A B,$ and (d) $B A.$$$A=\left[\begin{array}{cc}10 & 7 \\-4 & 6\end{array}\right], B=\left[\begin{array}{cc}0 & -12 \\8 & 11\end{array}\right]$$
Find, if possible, (a) $A-B,$ (b) $2 B-3 A,$ (c) $A B,$ and (d) $B A.$$$A=\left[\begin{array}{rrr}-2 & -3 & 6 \\4 & 5 & 7 \\1 & 7 & 4\end{array}\right], B=\left[\begin{array}{lll}1 & 4 & 2 \\0 & 1 & 6 \\0 & 3 & 1\end{array}\right]$$
Find, if possible, (a) $A-B,$ (b) $2 B-3 A,$ (c) $A B,$ and (d) $B A.$$$A=\left[\begin{array}{rr}-1 & 4 \\5 & 1 \\0 & -1\end{array}\right], B=\left[\begin{array}{lll}0 & 4 & 0 \\3 & 1 & -2\end{array}\right]$$