Chapter 11

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

Find the \(n\) th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10 th term? $$a=\frac{5}{2}, d=-\frac{1}{2}$$

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

Use Pascal's triangle to expand the expression. $$\left(x^{2} y-1\right)^{5}$$

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=\frac{5}{2}, \quad r=-\frac{1}{2}$$

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

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

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

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

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

Use Pascal's triangle to expand the expression. $$(1+\sqrt{2})^{6}$$

Annuity What is the present value of an annuity that consists of 20 semiannual payments of \(\$ 1000\) at the interest rate of \(9 \%\) per year, compounded semiannually?

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

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$5,8,11,14, \dots$$

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

Determine whether the sequence is geometric. If it is geometric, find the common ratio. $$2,4,8,16, \dots$$

Use Pascal's triangle to expand the expression. $$(2 x-3 y)^{3}$$