Chapter 12

$\mathrm{-1},2,5,8,11$

$\mathrm{-3},\mathrm{-1},1,3,5$

$7,13,31,85,247$

$\mathrm{-3},\mathrm{-1},3,11,27$

$\mathrm{-1},4,\mathrm{-9},16,\mathrm{-25}$

$1,\mathrm{-8},27,\mathrm{-64},125$

$a_n=3n$

$a_n=5n$

$a_n={\left(\mathrm{-1}\right)}^n{3}^n$

$a_n={\left(\mathrm{-1}\right)}^{n+1}{n}^2$

$a_n=\frac{1}{2^n}$

$a_n=\frac{1}{n^2}$

$2,1,\frac{1}{3},\frac{1}{12},\frac{1}{60}$

$3,\frac{3}{2},\frac{1}{2},\frac{1}{8},\frac{1}{40}$

$\frac{1}{2},\frac{1}{6},\frac{1}{12},\frac{1}{20},\frac{1}{30}$

$\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}$

45

60

$\frac{16}{3}$

8

${\displaystyle \sum _{n=1}^5\frac{1}{2^n}}$

${\displaystyle \sum _{n=1}^5\frac{1}{n^2}}$

${\displaystyle \sum _{n=1}^5{(\mathrm{-1})}^{n+1}{n}^2}$

${\displaystyle \sum _{n=1}^5{(\mathrm{-1})}^{n}2n}$

ⓐ The sequence is arithmetic with common difference $d=11$. ⓑ The sequence is arithmetic with common difference $d=\mathrm{-6}$.
ⓒ The sequence is not arithmetic as all the differences between the consecutive terms are not the same.

ⓐ The sequence is not arithmetic as all the differences between the consecutive terms are not the same. ⓑ The sequence is arithmetic with common difference $d=2.$
ⓒ The sequence is arithmetic with common difference $d=\mathrm{-5}.$

$7,3,\mathrm{-1},\mathrm{-5},\mathrm{-9}\text{,}\text{…}$

$11,3,\mathrm{-5},\mathrm{-13},\mathrm{-21}\text{,}\text{…}$

241

$\mathrm{-106}$

$a_{11}=2.$ The general term is $a_n=\mathrm{-3}n+35.$

$a_{19}=\mathrm{-55}.$ The general term is $a_n=\mathrm{-4}n+21.$

$a_1=5,$$d=4.$ The general term is $a_n=4n+1.$

$a_1=8,$$d=\mathrm{-3}.$ The general term is $a_n=\mathrm{-3}n+11.$

1,890

1,515

2,300

5,250

2,670

3,045

ⓐ The sequence is geometric with common ratio $r=3.$ ⓑ The sequence is geometric with common ratio $d=\frac{1}{4}.$ ⓒ The sequence is not geometric. There is no common ratio.

ⓐ The sequence is not geometric. There is no common ratio. ⓑ The sequence is geometric with common ratio $r=2.$ ⓒ The sequence is geometric with common ratio $r=\frac{1}{2}.$

$7,\mathrm{-21},63,\mathrm{-189},567$

$6,\mathrm{-24},96,\mathrm{-384},1536$

$\frac{1}{\mathrm{6,561}}$

$\frac{1}{\mathrm{16,384}}$

$a_9=\mathrm{39,366}.$ The general term is $a_n=6{(3)}^{n-1}.$

$a_{11}=\mathrm{7,168}.$ The general term is $a_n=7{(2)}^{n-1}.$

$\mathrm{3,145,725}$

$\mathrm{10,460,353,200}$

$\mathrm{393,204}$

$\mathrm{10,230}$

96

$\frac{256}{3}$

$\frac{4}{9}$

$\frac{8}{9}$

$\text{\$}\mathrm{10,000}$

$\text{\$}3,333.33$

$\text{\$}\mathrm{88,868.36}$

$\text{\$}\mathrm{698,201.57}$

$x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^3$
$+5x{y}^4+y^5$

$p^7+7p^{6}q+21p^5{q}^2+35p^4{q}^3$
$+35p^3{q}^4+21p^2{q}^5+7p{q}^6+q^7$

$x^4+8x^3+24x^2+32x+16$

$x^6+6x^5+15x^4+20x^3+15x^2$
$+6x+1$

$16x^4-96x^3+216x^2-216x+81$

$64x^6-192x^5+240x^4-160x^3$
$+60x^2-12x+1$

ⓐ 6 ⓑ 1 ⓒ 1 ⓓ 35

ⓐ 2 ⓑ 1 ⓒ 1 ⓓ 6

$x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^3$
$+5x{y}^4+y^5$

$m^6+6m^{5}n+15m^4{n}^2+20m^3{n}^3$
$+15m^2{n}^4+6m{n}^5+n^6$

$x^5-15x^4+90x^3-270x^2$
$+405x-243$

$y^6-6y^5+15y^4-20y^3+15y^2$
$-6y+1$

$243x^5-810x^{4}y+1080x^3{y}^2$
$-720x^2{y}^3+240x{y}^4-32y^5$

$256x^4-768x^{3}y+864x^2{y}^2$
$-432x{y}^3+81y^4$

$15x^4{y}^2$

$8a{b}^7$

7,168

280

$\mathrm{-5},\mathrm{-3},\mathrm{-1},1,3$

$4,7,10,13,16$

$5,7,11,19,35$

$1,5,21,73,233$

$2,1,\frac{8}{9},1,\frac{32}{25}$

$1,\frac{3}{2},\frac{5}{4},\frac{7}{8},\frac{9}{16}$

$\mathrm{-2},4,\mathrm{-6},8,\mathrm{-10}$

$1,\mathrm{-4},9,\mathrm{-16},25$

$1,-\frac{1}{4},\frac{1}{9},-\frac{1}{16},\frac{1}{25}$

$a_n=8n$

$a_n=n+5$

$a_n=e^{n+2}$

$a_n={(\mathrm{-1})}^{n}5n$

$a_n={(\mathrm{-1})}^n{n}^3$

$a_n={(\mathrm{-1})}^{n}2n$

$a_n=\frac{1}{4^n}$

$a_n=-\frac{n}{n+1}$

$-\frac{5}{2^n}$

$4,2,\frac{2}{3},\frac{1}{6},\frac{1}{30}$

$3,6,18,72,360$

$2,24,720,40320,3628800$

$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}$

$1,\frac{1}{2},\frac{2}{3},\frac{3}{2},\frac{24}{5}$

$2,\frac{3}{2},\frac{8}{3},\frac{15}{2},\frac{144}{5}$

$1+4+9+16+25=55$

$5+7+9+11+13+15=60$

$2+4+8+16=30$

$\frac{4}{1}+\frac{4}{1}+\frac{4}{2}+\frac{4}{6}+\frac{32}{3}=10\frac{2}{3}$

$2+6+12+20+30=70$

$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}=\frac{71}{20}$

${\displaystyle \sum _{n=1}^5\frac{1}{3^n}}$

${\displaystyle \sum _{n=1}^5\frac{1}{n^3}}$

${\displaystyle \sum _{n=1}^5\frac{2}{n}}$

${\displaystyle \sum _{n=1}^5{(\mathrm{-1})}^{n+1}3n}$

${\displaystyle \sum _{n=1}^{10}{(\mathrm{-1})}^{n}2n}$

${\displaystyle \sum _{n=1}^7(2n+12)}$

Answers will vary.

Answers will vary.

The sequence is arithmetic with common difference $d=8.$

The sequence is not arithmetic.

The sequence is arithmetic with common difference $d=\mathrm{-3}.$

$11,18,25,32,39$

$\mathrm{-7},\mathrm{-3},1,5,9$

$14,5,\mathrm{-4},\mathrm{-13},\mathrm{-22}$

$163$

$131$

$\mathrm{-79}$

$a_{20}=\mathrm{-34}.$ The general term is $a_n=\mathrm{-2}n+6.$

$a_{11}=59.$ The general term is $a_n=5n+4.$

$a_8=\mathrm{-13}.$ The general term is $a_n=\mathrm{-5}n+27.$

$a_1=11,$$d=3.$ The general term is $a_n=3n+8.$

$a_1=21,$$d=\mathrm{-8}.$ The general term is $a_n=\mathrm{-8}n+29.$

$a_1=\mathrm{-15},$$d=3.$ The general term is $a_n=3n-18.$

1,635

$\mathrm{-1},065$

360

6,325

–3,575

6,280

4,125

$\mathrm{-3,850}$

Answers will vary.

Answers will vary.

The sequence is geometric with common ratio $r=4.$

The sequence is geometric with common ratio $r=\frac{1}{2}.$

The sequence is geometric with common ratio $r=\mathrm{-2}.$

The sequence is geometric with common ratio $r=\frac{1}{2}.$

The sequence is arithmetic with common difference $d=5.$

The sequence is geometric with common ratio $r=\frac{1}{2}.$

$4,12,36,108,324$

$\mathrm{-4},8,\mathrm{-16},32,\mathrm{-64}$

$27,9,3,1,\frac{1}{3}$

$\mathrm{472,392}$

3,072

$0.0001$

$a_9=\mathrm{2,304}.$ The general term is $a_n=9{(2)}^{n-1}.$

$a_{15}=-\frac{2}{19,683}.$ The general term is $a_n=\mathrm{-486}{\left(-\frac{1}{3}\right)}^{n-1}.$

$a_{10}=0.000000001.$ The general term is $a_n={(0.1)}^{n-1}.$

$\mathrm{57,395,624}$

$\mathrm{-65,538}$

$\frac{\mathrm{7,174,453}}{\mathrm{59,049}}\approx 121.5$

$\mathrm{65,534}$

$4088$

$\frac{\mathrm{29,524}}{6561}\approx 4.5$

$\frac{3}{2}$

$\frac{9}{2}$

no sum as $r\ge 1$

2,048

$\frac{1}{3}$

$\frac{7}{9}$

$\frac{5}{11}$

ⓐ $\text{\$}6666.67$ ⓑ $\text{\$}4000$ ⓒ $\text{\$}\mathrm{15,000}$ ⓒ $\text{\$}7500$

$\text{\$}\mathrm{295,581.88}$

$\text{\$}\mathrm{14,234.10}$

Answers will vary.

Answers will vary.

$a^8+8a^{7}b+28a^6{b}^2+56a^5{b}^3$
$+70a^4{b}^4+56a^3{b}^5+28a^2{b}^6$
$+8a{b}^7+b^8$

$p^9+9p^{8}q+36p^7{q}^2+84p^6{q}^3$
$+126p^5{q}^4+126p^4{q}^5+84p^3{q}^6$
$+36p^2{q}^7+9p{q}^8+q^9$

$a^6-6a^{5}b+15a^4{b}^2-20a^3{b}^3$
$+15a^2{b}^4-6a{b}^5+b^6$

$x^3+15x^2+75x+125$

$y^7+7y^6+21y^5+35y^4+35y^3$
$+21y^2+7y+1$

$z^6-12z^5+60z^4-160z^3+240z^2$
$-192z+64$

$243x^5-405x^4+270x^3-90x^2$
$+15x-1$

$27x^3-135x^2+225x-125$

$27x^3+135x^{2}y+225x{y}^2+125y^3$

ⓐ 7 ⓑ 1 ⓒ 1 ⓓ 45

ⓐ 4 ⓑ 1 ⓒ 1 ⓓ 55

$m^5+5m^{4}n+10m^3{n}^2+10m^2{n}^3$
$+5m{n}^4+n^5$

$s^7+7s^{6}t+21s^5{t}^2+35s^4{t}^3$
$+35s^3{t}^4+21s^2{t}^5+7s{t}^6+t^7$

$y^4-12y^3+54y^2-108y+81$

$q^3-12q^2+48q-64$

$625x^4-1000x^{3}y+600x^2{y}^2$
$-160x{y}^3+16y^4$

$243x^5+1620x^{4}y+4320x^3{y}^2$
$+5760x^2{y}^3+3840x{y}^4+1024y^5$

$126a^5{b}^4$

$462x^5{y}^6$

112

324

30,618

Answers will vary.

Answers will vary.

$7,13,31,85,247$

$\frac{3}{4},\frac{5}{16},\frac{7}{64},\frac{9}{256},\frac{11}{1024}$

$a_n=9n$

$a_n=e^{n-4}$

$a_n=-\frac{n}{n+2}$

$\frac{1}{6},\frac{1}{12},\frac{1}{20},\frac{1}{30},\frac{1}{42}$

$\mathrm{-3}+(\mathrm{-1})+1+3+5$
$+7+9=21$

$4+4+2+\frac{2}{3}+\frac{1}{6}=\frac{65}{6}$

${\displaystyle \sum _{n=1}^5{(\mathrm{-1})}^n\frac{1}{3^n}}$

${\displaystyle \sum _{n=1}^5\frac{4}{n}}$

The sequence is arithmetic with common difference $d=6.$

$5,8,11,14,17$

$\mathrm{-13},\mathrm{-7},\mathrm{-1},5,11$

$\mathrm{-129}$

$a_{18}=103.$ The general term is $a_n=7n-23.$

$a_1=1,$$d=4.$ The general term is $a_n=4n-3.$

$\mathrm{-430}$

$585$

$4850$

$980$

The sequence is not geometric.

The sequence is geometric with common ratio $r=\mathrm{-2}.$

$128,32,8,2,\frac{1}{2}$

$\mathrm{1,536}$

$a_{12}=\mathrm{-25},165,824.$ The general term is $a_n=6{(\mathrm{-4})}^{n-1}.$

5,460

$\approx 3906.25$

$\frac{189}{8}=23.625$

$\frac{343}{6}\approx 57.167$

$\frac{4}{11}$

$\text{\$}\mathrm{1,634,421.27}$

$x^4-4x^{3}y+6x^2{y}^2-4x{y}^3+y^4$

$32y^5-240y^4+720y^3-1080y^2$
$+810y-243$