Chapter 20

Evaluate each factorial. 6 !

Evaluate each limit. $$\lim _{b \rightarrow 0}(b-c+5)$$

Find the fifth term of a GP with first term 5 and common ratio 2

Find the fifteenth term of an AP with first term 4 and common difference 3

Do not simplify or give the decimal value of any fractions in this exercise. Write the first five terms of each series, given the general term. $$u_{n}=3 n$$

Evaluate each factorial. 8 !

Evaluate each limit. $$\lim _{b \rightarrow 0}\left(a+b^{2}\right)$$

Find the fourth term of a GP with first term 7 and common ratio -4

Find the tenth term of an AP with first term 8 and common difference 2

Do not simplify or give the decimal value of any fractions in this exercise. Write the first five terms of each series, given the general term. $$u_{n}=2 n+3$$

Evaluate each factorial. \(\frac{7 !}{5 !}\)

Find the sixth term of a GP with first term -3 and common ratio 5

Find the twelfth term of an AP with first term -1 and common difference 4

Do not simplify or give the decimal value of any fractions in this exercise. Write the first five terms of each series, given the general term. $$u_{n}=\frac{n+1}{2}$$

Evaluate each factorial. \(\frac{11 !}{9 ! 2 !}\)

Evaluate each limit. $$\lim _{h \rightarrow 0} \frac{a+b+c}{b+c-5}$$

Find the fifth term of a GP with first term -4 and common ratio -2

Find the ninth term of an AP with first term -5 and common difference -2

Do not simplify or give the decimal value of any fractions in this exercise. Write the first five terms of each series, given the general term. $$u_{n}=\frac{2^{n}}{n}$$

Evaluate each factorial. \(\frac{7 !}{3 ! 4 !}\)

Find the sum of the infinitely many terms of each GP. $$144,72,36,18, \dots$$

Find the eleventh term of the AP \(9,13,17, \dots\)

Deduce the general term of each series. Use it to predict the next two terms. $$2+4+6+\cdots$$

Evaluate each factorial. \(\frac{8 !}{3 ! 5 !}\)

Find the sum of the infinitely many terms of each GP. $$8,4,2,1, \frac{1}{2}, \dots$$

Find the eighth term of the AP $$-5,-8,-11, \ldots$$

Deduce the general term of each series. Use it to predict the next two terms. $$1+8+27+\cdots$$

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$(x+y)^{7}=x^{7}+7 x^{6} y+21 x^{5} y^{2}+35 x^{4} y^{3}+35 x^{3} y^{4}+21 x^{2} y^{5}+7 x y^{6}+y^{7}$$

Find the sum of the infinitely many terms of each GP. $$10,2,0.4,0.08, \ldots$$

Find the ninth term of the AP $$x, \quad x+3 y, \quad x+6 y, \ldots$$

Deduce the general term of each series. Use it to predict the next two terms. $$\frac{2}{4}+\frac{4}{5}+\frac{8}{6}+\frac{16}{7}+\cdots$$

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$(4+3 b)^{4}=256+768 b+864 b^{2}+432 b^{3}+81 b^{4}$$

Find the sum of the infinitely many terms of each GP. $$1, \frac{1}{4}, \frac{1}{16}, \dots$$

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$(3 a-2 b)^{4}=81 a^{4}-216 a^{3} b+216 a^{2} b^{2}-96 a b^{3}+16 b^{4}$$

Insert a geometric mean between 5 and \(45 .\)

Write the first five terms of each AP. First term is 3 and thirteenth term is 55

Deduce a recursion relation for each series. Use it to predict the next two terms. $$1+5+9+\dots$$

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(x^{3}+y\right)^{7}=x^{21}+7 x^{18} y+21 x^{15} y^{2}+35 x^{12} y^{3}+35 x^{9} y^{4}+21 x^{6} y^{5}+7 x^{3} y^{6}+y^{7}$$

Insert a geometric mean between 7 and \(112 .\)

Write the first five terms of each AP. First term is 5 and tenth term is 32

Deduce a recursion relation for each series. Use it to predict the next two terms. $$5+15+45+\cdots$$

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(x^{1 / 2}+y^{2 / 3}\right)^{5}=x^{5 / 2}+5 x^{2} y^{2 / 3}+10 x^{3 / 2} y^{4 / 3}+10 x y^{2}+5 x^{1 / 2} y^{8 / 3}+y^{10 / 3}$$

"A runner can never reach a finish line 1 mile away because first he would have to run half a mile, and then must run half of the remaining distance, or \(\frac{1}{4}\) mile, and then half of that, and so on. Since there are an infinite number of distances that must be run, it will take an infinite length of time, and so the runner will never reach the finish line." Show that the distances to be run form the infinite series; $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots $$ Then disprove the paradox by actually finding the sum of that series and showing that the sum is not infinite.

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(1 / x+1 / y^{2}\right)^{3}=1 / x^{3}+3 / x^{2} y^{2}+3 / x y^{4}+1 / y^{6}$$

Insert a geometric mean between -21 and -84

Insert two geometric means between 8 and 216

Find the first term of an AP whose common difference is 3 and whose seventh term is 11

Writing: State in your own words the difference between an arithmetic progression and a geometric progression. Give a real-world example of each.

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(1 / \sqrt{x}+y^{2}\right)^{4}=x^{-2}+4 x^{-3 / 2} y^{2}+6 x^{-1} y^{4}+4 x^{-1 / 2} y^{6}+y^{8}$$

Insert two geometric means between 9 and -243