Chapter 12

An arithmetic sequence is a sequence in which the ____ between successive terms is constant.

A sequence is a function whose domain is ____________.

An annuity is a sum of money that is paid in regular equal payments. The _____ of an annuity is the sum of all the individual payments together with all the interest.

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.

The \(n\) th partial sum of a sequence is the sum of the first __________ terms of the sequence. So for the sequence \(a_{n}=n^{2}\) the fourth partial sum is \(S_{4}=\) _____ \(+\) ______ \(+\) ______ \(+\) _____ \(=\) _____ .

The sequence given by \(a_{n}=a+(n-1) d\) 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 sequence given by \(a_{n}=a r^{n-1}\) 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 __________.

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."

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=n-3\)

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

True or False? If False, give a reason. The \(n\) th partial sum of an arithmetic sequence is the average of the first and last terms times \(n .\)

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

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

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=2 n-1\)

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.

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

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

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=\frac{1}{2 n+1}\)

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.

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). $$a_{n}=7(3)^{n-1}$$

The \(n\) th term of an arithmetic 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 part (a). $$a_{n}=7+3(n-1)$$

Pascal's Triangle 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 \(n\) $$5+8+11+\dots+(3 n+2)=\frac{n(3 n+7)}{2}$$

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=n^{2}-1\)

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

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.

The \(n\) th term of an arithmetic 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 part (a). $$a_{n}=-10+20(n-1)$$

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). $$a_{n}=6(-0.5)^{n-1}$$

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

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=5^{n}\)

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.

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). $$a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1}$$

The \(n\) th term of an arithmetic 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 part (a). $$a_{n}=-6-4(n-1)$$

Pascal's Triangle 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 \(n\) $$1 \cdot 2+2 \cdot 3+3 \cdot 4+\dots+n(n+1)=\frac{n(n+1)(n+2)}{3}$$

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=\left(\frac{-1}{3}\right)^{n}\)

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.

The \(n\) th term of an arithmetic 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 part (a). $$a_{n}=-10+4(n-1)$$

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). $$a_{n}=3^{n-1}$$

Pascal's Triangle 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 \(n\) $$1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\frac{n(n+1)(2 n+7)}{6}$$

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

The \(n\) th term of an arithmetic 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 part (a). $$a_{n}=\frac{5}{2}-(n-1)$$

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

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=7, \quad r=4$$

Pascal's Triangle 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 \(n\) $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$

The \(n\) th term of an arithmetic 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 part (a). $$a_{n}=\frac{1}{2}(n-1)$$

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