Chapter 12

\(1-12\) . 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.

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

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

1–4 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. \(a_{n}=n+1\)

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (2 x+1)^{4} $$

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.

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

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

1–4 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. \(a_{n}=2 n+3\)

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(x+\frac{1}{x}\right)^{4} $$

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.

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

1–4 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. \(a_{n}=\frac{1}{n+1}\)

\(1-12\) . 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.

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

1–4 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. \(a_{n}=n^{2}+1\)

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (x-1)^{5} $$

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.

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

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

5-8 Find the \(n\)th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10th term? $$a=3, d=5$$

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

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (\sqrt{a}+\sqrt{b})^{6} $$

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

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

5-8 Find the \(n\)th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10th term? $$a=-6, d=3$$

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

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(x^{2} y-1\right)^{5} $$

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

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? $$ a=\frac{5}{2}, \quad r=-\frac{1}{2} $$

5-8 Find the \(n\)th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10th term? $$a=\frac{5}{2}, d=-\frac{1}{2}$$

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

\(1-12\) . 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 semiannully?

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? $$ a=\sqrt{3}, \quad r=\sqrt{3} $$

5-8 Find the \(n\)th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10th term? $$a=\sqrt{3}, d=\sqrt{3}$$

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

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (2 x-3 y)^{3} $$

Funding an Annuity How much money must be invested now at 9\(\%\) per year, compounded semiannully, to fund an annuity of 20 payments of \(\$ 200\) each, paid every 6 months, the first payment being 6 months from now?