Chapter 12

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.

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

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 ____________.

The ____________ ____________ of an annuity is the amount that must be invested now at interest rate \(i\) per time period to provide \(n\) payments each of amount \(R\)

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 _____.

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.

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

Find the first four terms and the 100th term of the sequence. $$a_{n}=n+1$$

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

Find the amount of an annuity that consists of24 monthly payments of \(\$ 500\) each into an account that pays \(8 \%\) interest per year, compounded monthly.

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

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

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

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). $$a_{n}=5(2)^{n-1}$$

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.

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). $$a_{n}=5+2(n-1)$$

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

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). $$a_{n}=3(-4)^{n-1}$$

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.

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). $$a_{n}=3-4(n-1)$$

Find the first four terms and the 100th term of the sequence. $$a_{n}=n^{2}+1$$

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

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

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.

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

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

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

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.

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

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

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). $$a_{n}=\frac{1}{2}(n-1)$$

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

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

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

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? $$a=3, \quad r=5$$

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

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

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

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=-6, \quad r=3$$

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