Chapter 14

Fill in each blank with the correct response. In each row of Pascal's triangle, the first and last terms are __________ , and each number in the interior of the triangle is the _________ of the two numbers just above it (one to the right and one to the left).

Fill in each blank with the correct response. In a geometric sequence, if any term after the first is divided by the term that precedes it, the result is the common _______ of the sequence.

Fill in each blank with the correct response. For the geometric sequence having \(a_{n}=(-2)^{n},\) the term \(a_{5}=\) _____.

Fill in each blank with the correct response. In the sequence \(3,6,9,12,\) the term \(a_{3}=\) _____.

Fill in each blank with the correct response. The sum of the first five terms of the geometric sequence \(1,2,4, \ldots\) is _____.

Fill in each blank with the correct response. If \(a_{n}=2 n,\) then \(a_{40}=\) ____.

Fill in each blank with the correct response. If \(a_{n}=(-1)^{n},\) then \(a_{115}=\) ____.

Fill in each blank with the correct response. The number of terms in the geometric sequence \(1,2,4, \ldots, 2048\) is ______.

Fill in each blank with the correct response. The value of the sum \(\sum_{i=1}^{3}(i+2)\) is ____.

Fill in the blanks to complete the terms of each geometric sequence. \(\frac{1}{3},-\frac{1}{9}, \frac{1}{27}\) _____,_____,_____.

Fill in each blank with the correct response. The value of \(0 !\) is _______.

Fill in each blank with the correct response. The arithmetic mean of \(-4,-2,0,2,\) and 4 is ____.

Fill in each blank with the correct response. For any nonnegative integer \(n,\) the binomial coefficient \({ }_{n} C_{0}\) is equal to ______.

Write the first five terms of each sequence. $$ a_{n}=n+1 $$

Fill in each blank with the correct response. For any nonnegative integer \(n,\) the binomial coefficient \({ }_{n} C_{n}\) is equal to ______.

Write the first five terms of each sequence. $$ a_{n}=n+4 $$

Evaluate each expression. $$ 6 ! $$

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(1,2,3,4,5, \ldots\)

Write the first five terms of each sequence. $$ a_{n}=\frac{n+3}{n} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 4,8,16,32, \ldots $$

Evaluate each expression. $$ 4 ! $$

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(2,5,8,11, \ldots\)

Write the first five terms of each sequence. $$ a_{n}=\frac{n+2}{n} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 5,15,45,135, \ldots $$

Evaluate each expression. $$ 8 ! $$

Write the first five terms of each sequence. $$ a_{n}=3^{n} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \ldots $$

Evaluate each expression. $$ 9 ! $$

Write the first five terms of each sequence. $$ a_{n}=2^{n} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ \frac{5}{7}, \frac{8}{7}, \frac{11}{7}, 2, \ldots $$

Evaluate each expression. $$ \frac{6 !}{4 ! 2 !} $$

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(10,5,0,-5,-10, \ldots\)

Write the first five terms of each sequence. $$ a_{n}=-\frac{1}{n^{2}} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 1,-3,9,-27,81, \ldots $$

Evaluate each expression. $$ \frac{7 !}{3 ! 4 !} $$

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(-6,-10,-14,-18, \ldots\)

Write the first five terms of each sequence. $$ a_{n}=-\frac{2}{n^{2}} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 2,-8,32,-128, \ldots $$

Write the first five terms of each arithmetic sequence. \(a_{1}=5, d=4\)

Write the first five terms of each sequence. $$ a_{n}=5(-1)^{n-1} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \ldots $$

Evaluate each expression. $$ \frac{4 !}{0 ! 4 !} $$

Write the first five terms of each arithmetic sequence. \(a_{1}=6, d=7\)

Write the first five terms of each sequence. $$ a_{n}=6(-1)^{n+1} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ \frac{2}{3},-\frac{2}{15}, \frac{2}{75},-\frac{2}{375}, \ldots $$

Evaluate each expression. $$ \frac{5 !}{5 ! 0 !} $$

Write the first five terms of each arithmetic sequence. \(a_{1}=-2, d=-4\)

Write the first five terms of each sequence. $$ a_{n}=n-\frac{1}{n} $$

Evaluate each expression. $$ 4 ! \cdot 5 $$

Write the first five terms of each arithmetic sequence. \(a_{1}=-3, d=-5\)