Chapter 13

An algebraic expression of the form \(a+b,\) which consists of a sum of two terms, is called a _______

A geometric sequence is a sequence in which the _____ of successive terms is constant.

A sequence is a function whose domain is _____.

The sequence \(a_{n}=a+(n-1) d\) is an arithmetic sequence in which a is the first term and \(d\) is the ________ __________ So for the arithmetic sequence \(a_{n}=2+5(n-1)\) the first term is ___________and the common difference is _____________

Which of the following is true about Step 2 in a proof by mathematical induction? (i) We prove "P \((k+1)\) is true." (ii) We prove "If \(P(k)\) is true, then \(P(k+1)\) is true."

The sequence \(a_{n}=a r^{n-1}\) is a geometric sequence in which a is the first term and \(r\) is the _____ _____. So for the geometric sequence \(a_{n}=2(5)^{n-1}\) the first term is _____ and the common ratio is _____.

True or false? The \(n\) th partial sum of an arithmetic sequence is the average of the first and last terms times \(n .\)

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=n+1 $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 2+4+6+\cdots+2 n=n(n+1) $$

True or false? If we know the first and second terms of a geometric sequence, then we can find any other term.

Annuity Find the amount of an annuity that consists of 10 annual payments of \(\$ 1000\) each into an account that pays 6\(\%\) interest per year.

True or false? If we know the first and second terms of an arithmetic sequence, then we can find any other term.

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=2 n+3 $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1+4+7+\cdots+(3 n-2)=\frac{n(3 n-1)}{2} $$

Annuity Find the amount of an annuity that consists of 24 monthly payments of \(\$ 500\) each into an account that pays 8\(\%\) interest per year, compounded monthly.

A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$ a_{n}=5+2(n-1) $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=\frac{1}{n+1} $$

Use Pascal’s triangle to expand the expression. $$ (x+y)^{6} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 5+8+11+\cdots+(3 n+2)=\frac{n(3 n+7)}{2} $$

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=5(2)^{n-1} $$

Annuity Find the amount of an annuity that consists of 20 annual payments of \(\$ 5000\) each into an account that pays interest of 12\(\%\) per year.

\(5-8=\) A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$ a_{n}=3-4(n-1) $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=n^{2}+1 $$

Use Pascal’s triangle to expand the expression. $$ (2 x+1)^{4} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6} $$

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=3(-4)^{n-1} $$

Annuity Find the amount of an annuity that consists of 20 semiannual payments of \(\$ 500\) each into an account that pays 6\(\%\) interest per year, compounded semiannually.

\(5-8=\) A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$ a_{n}=\frac{5}{2}-(n-1) $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=\frac{(-1)^{n}}{n^{2}} $$

Use Pascal’s triangle to expand the expression. $$ \left(x+\frac{1}{x}\right)^{4} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{n(n+1)(n+2)}{3} $$

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1} $$

Annuity Find the amount of an annuity that consists of 16 quarterly payments of \(\$ 300\) each into an account that pays 8\(\%\) interest per year, compounded quarterly.

\(5-8=\) A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$ a_{n}=\frac{1}{2}(n-1) $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=\frac{1}{n^{2}} $$

Use Pascal’s triangle to expand the expression. $$ (x-y)^{5} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\frac{n(n+1)(2 n+7)}{6} $$

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=3^{n-1} $$

Annuity Find the amount of an annuity that consists of 40 annual payments of \(\$ 2000\) each into an account that pays interest of 5\(\%\) per year.

\(9-12\) . Find the \(n\) th term of the arithmetic sequence with given first term and common difference \(d\) What is the 10 the term? $$ a=3, d=5 $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=1+(-1)^{n} $$

Use Pascal’s triangle to expand the expression. $$ (x-1)^{5} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$ a=3, \quad r=5 $$

Saving How much money should be invested every quarter at 10\(\%\) per year, compounded quarterly, to have \(\$ 5000\) in 2 years?

\(9-12\) . Find the \(n\) th term of the arithmetic sequence with given first term and common difference \(d\) What is the 10 the term? $$ a=-6, d=3 $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=(-1)^{n+1} \frac{n}{n+1} $$

Use Pascal’s triangle to expand the expression. $$ (\sqrt{a}+\sqrt{b})^{6} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1^{3}+3^{3}+5^{3}+\cdots+(2 n-1)^{3}=n^{2}\left(2 n^{2}-1\right) $$

Saving How much money should be invested monthly at 6\(\%\) per year, compounded monthly, to have \(\$ 2000\) in 8 months?