Write the first five terms of the geometric sequence. $a_{1}=\frac{1}{4}, r=7 x$

Write the first five terms of the geometric sequence. $a_{1}=3, r=\frac{x}{2}$

Find the indicated partial sum of the series. $\sum_{i=1}^{\infty} \frac{2}{3^{i}}$ Fifth partial sum

Consider n independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is p and the probability of a failure is q = 1 - p. In this context, the term $_{n} C_{k} p^{k} q^{n-k}$ in the expansion of $(p+q)^{n}$ gives the probability of k successes in the n trials of the experiment. The probability of a baseball player getting a hit during any given time at bat is $\frac{1}{4}$. To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term $_{10} \mathrm{C}_{3}\left(\frac{1}{4}\right)^{3}\left(\frac{3}{4}\right)^{7}$ in the expansion of $\left(\frac{1}{4}+\frac{3}{4}\right)^{10}$

Determine whether the statement is true or false. Justify your answer. You can find the nth term of a geometric sequence by multiplying its common ratio by the first term of the sequence raised to the (n − 1)th power.

Find the indicated partial sum of the series. $\sum_{i=1}^{\infty} \frac{7}{5^{i}}$ Fourth partial sum

Consider n independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is p and the probability of a failure is q = 1 - p. In this context, the term $_{n} C_{k} p^{k} q^{n-k}$ in the expansion of $(p+q)^{n}$ gives the probability of k successes in the n trials of the experiment. A fair coin is tossed seven times. To find the probability of obtaining four heads, evaluate the term $_{7} C_{4}\left(\frac{1}{2}\right)^{4}\left(\frac{1}{2}\right)^{3}$ in the expansion of $\left(\frac{1}{2}+\frac{1}{2}\right)^{7}$.

Determine whether the statement is true or false. Justify your answer. The first n terms of a geometric sequence with a common ratio of 1 are the same as the first n terms of an arithmetic sequence with a common difference of 0 if both sequences have the same first term.

Use a graphing utility to graph the functions in the given order and in the same viewing window. Compare the graphs. Which two functions have identical graphs, and why? (a) $f(x)=\left(1-\frac{1}{2} x\right)^{4}$ (b) $g(x)=1-2 x+\frac{3}{2} x^{2}$ (c) $h(x)=1-2 x+\frac{3}{2} x^{2}-\frac{1}{2} x^{3}$ (d) $p(x)=1-2 x+\frac{3}{2} x^{2}-\frac{1}{2} x^{3}+\frac{1}{16} x^{4}$

Determine whether the statement is true or false. Justify your answer. For a geometric sequence in which the quotient of term $a_{7}$ and term $a_{5}$ is 16, the common ratio is 8.

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $\frac{1}{2}+\frac{2}{4}+\frac{6}{8}+\frac{24}{16}+\frac{120}{32}+\frac{720}{64}$

Write the first five terms of the sequence. (Assume n begins with 1.) $a_{n}=\left(-\frac{1}{2}\right)^{n}$

The manufacturer of a new food processor plans to produce and sell 8000 units per year. Each year, $10 of all units sold become inoperative. So, 8000 units will be in use after 1 year,

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$