1.

1. [latex]P_0=20.[/latex] [latex]P_n=P_{n−1}+5[/latex]
2. [latex]P_n=20+5n[/latex]

3.

1. [latex]P_1=P_0+15=40+15=55.[/latex] [latex]P_2=55+15=70[/latex]
2. [latex]P_n=40+15n[/latex]
3. [latex]P_{10}=40+15(10)=190[/latex] thousand dollars
4. [latex]40+15n=100[/latex] when [latex]n=4[/latex] years.

5. Grew [latex]64[/latex] in [latex]8[/latex] weeks: [latex]8[/latex] per week

1. [latex]P_n=3+8n[/latex]
2. [latex]187=3+8n.[/latex] [latex]n=23[/latex] weeks

7.

1. [latex]P_0=200[/latex](thousand), [latex]P_n=(1+.09)[latex]P_{n−1}[/latex] where [latex]n[/latex] is years after 2000
2. [latex]P_n=200(1.09)^n[/latex]
3. [latex]P_{16}=200(1.09)^{16}=794.061[/latex](thousand)[latex]=794,061[/latex]
4. [latex]200(1.09)^n=400. n=log(2)/log(1.09)=8.043[/latex]. In 2008

9. Let [latex]n=0[/latex] be 1983. [latex]P_n=1700(2.9)^n[/latex]. 2005 is [latex]n=22[/latex]. [latex]P_{22}=1700(2.9)^{22}=25,304,914,552,324[/latex] people. Clearly not realistic, but mathematically accurate.

11. If [latex]n[/latex] is in hours, better to start with the explicit form. [latex]P_0=300.[/latex] [latex]P_4=500=300(1+r)^4[/latex]
[latex]500/300=(1+r)^4.[/latex]
[latex]1+r=1.136.[/latex]
[latex]r=0.136[/latex]

1. [latex]P_0=300. P_n=(1.136)P_{n−1}[/latex]
2. [latex]P_n=300(1.136)^n[/latex]
3. [latex]P_{24}=300(1.136)^24=6400[/latex] bacteria
4. [latex]300(1.136)n=900.n=log(3)/log(1.136)=[/latex] about [latex]8.62[/latex] hours

13.

1. [latex]P_0=100[/latex] [latex]P_n=P_{n−1}+0.70(1−P_{n−1}/2000)P_{n−1}[/latex]
2. [latex]P_1=100+0.70(1−100/2000)(100)=166.5[/latex]
3. [latex]P_2=166.5+0.70(1−166.5/2000)(166.5)=273.3[/latex]

15. To find the growth rate, suppose [latex]n=0[/latex] was [latex]1968[/latex]. Then [latex]P_0[/latex] would be [latex]1.60[/latex] and [latex]P2=2.30=1.60(1+r)^8,r=0.0464[/latex]. Since we want [latex]n=0[/latex] to correspond to 1960, then we don't know [latex]P_0[/latex], but [latex]P_8[/latex] would be [latex]1.60=P_0(1.0464)^8[/latex]. [latex]P_0=1.113[/latex]

1. [latex]P_n=1.113(1.0464)^n[/latex]
2. [latex]P_0=$1.113[/latex], or about [latex]$1.11[/latex]
3. 1996 would be [latex]n=36.[/latex] [latex]P_36=1.113(1.0464)^36=$5.697[/latex]. Actual is slightly lower.

17. The population in the town was [latex]4000[/latex] in 2005, and is growing by [latex]4\%[/latex] per year.