Chapter 13

The 20 th term of an arithmetic sequence is \(101,\) and the common difference is \(3 .\) Find a formula for the nth term.

The common ratio in a geometric sequence is \(\frac{3}{2},\) and the fifth term is \(1 .\) Find the first three terms.

\(37-40\) . Find the first four partial sums and the nth partial sum of the sequence \(a_{m}\) \(a_{n}=\log \left(\frac{n}{n+1}\right) \quad[\text { Hint: Use a property of logarithms to }\) write the \(n\) th term as a difference.]

Find the term containing \(y^{3}\) in the expansion of \((\sqrt{2}+y)^{12}\)

Which term of the arithmetic sequence \(1,4,7, \ldots\) is \(88 ?\)

Which term of the geometric sequence \(2,6,18, \ldots\) is \(118,098 ?\)

\(41-48\) Find the sum. $$ \sum_{k=1}^{4} k $$

Find the term containing \(b^{8}\) in the expansion of \(\left(a+b^{2}\right)^{12}\)

The first term of an arithmetic sequence is \(1,\) and the common difference is \(4 .\) Is \(11,937\) a term of this sequence? If so, which term is it?

The second and fifth terms of a geometric sequence are 10 and \(1250,\) respectively. Is \(31,250\) a term of this sequence? If so, which term is it?

\(41-48\) Find the sum. $$ \sum_{k=1}^{4} k^{2} $$

Find the term that does not contain \(x\) in the expansion of $$ \left(8 x+\frac{1}{2 x}\right)^{8} $$

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=1, d=2, n=10 $$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a=5, \quad r=2, \quad n=6 $$

\(41-48\) Find the sum. $$ \sum_{k=1}^{3} \frac{1}{k} $$

Factor using the Binomial Theorem. $$ x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4} $$

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=3, d=2, n=12 $$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a=\frac{2}{3}, \quad r=\frac{1}{3}, \quad n=4 $$

\(41-48\) Find the sum. $$ \sum_{j=1}^{100}(-1)^{j} $$

Factor using the Binomial Theorem. $$ \begin{array}{l}{(x-1)^{5}+5(x-1)^{4}+10(x-1)^{3}} \\\ {+10(x-1)^{2}+5(x-1)+1}\end{array} $$

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=4, d=2, n=20 $$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a_{3}=28, \quad a_{6}=224, \quad n=6 $$

Factor using the Binomial Theorem $$ 8 a^{3}+12 a^{2} b+6 a b^{2}+b^{3} $$

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=100, d=-5, n=8 $$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a_{2}=0.12, \quad a_{5}=0.00096, \quad n=4 $$

\(41-48\) Find the sum. $$ \sum_{i=4}^{12} 10 $$

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a_{1}=55, d=12, n=10 $$

Find the sum. $$ 1+3+9+\cdots+2187 $$

\(41-48\) Find the sum. $$ \sum_{k=1}^{5} 2^{k-1} $$

Simplify using the Binomial Theorem. $$ \frac{(x+h)^{3}-x^{3}}{h} $$

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a_{2}=8, a_{5}=9.5, n=15 $$

Find the sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{512} $$

\(41-48\) Find the sum. $$ \sum_{i=1}^{3} i 2^{i} $$

Simplify using the Binomial Theorem. $$ \frac{(x+h)^{4}-x^{4}}{h} $$

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ 1+5+9+\cdots+401 $$

Find the sum. $$ \sum_{k=0}^{10} 3\left(\frac{1}{2}\right)^{k} $$

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{k=1}^{10} k^{2} $$

Simplify using the Binomial Theorem. Show that \((1.01)^{100}>2 .\) [Hint: Note that \((1.01)^{100}=(1+0.01)^{100},\) and use the Binomial Theorem to show that the sum of the first two terms of the expansion is greater than \(2 . ]\)

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ -3+\left(-\frac{3}{2}\right)+0+\frac{3}{2}+3+\cdots+30 $$

Find the sum. $$ \sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j} $$

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{k=1}^{100}(3 k+4) $$

Simplify using the Binomial Theorem. Show that \(\left(\begin{array}{l}{n} \\ {0}\end{array}\right)=1\) and \(\left(\begin{array}{l}{n} \\ {n}\end{array}\right)=1\)

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ 0.7+2.7+4.7+\cdots+56.7 $$

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots $$

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{j=7}^{20} j^{2}(1+j) $$

Simplify using the Binomial Theorem. Show that \(\left(\begin{array}{l}{n} \\\ {1}\end{array}\right)=\left(\begin{array}{c}{n} \\ {n-1}\end{array}\right)=n\)

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ -10-9.9-9.8-\cdots-0.1 $$

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots $$

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{j=5}^{15} \frac{1}{j^{2}+1} $$

Simplify using the Binomial Theorem. Show that \(\left(\begin{array}{l}{n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{n} \\ {n r}\end{array}\right) \quad\) for \(0 \leq r \leq n\)