Answer:
1. Juan's marathon training schedule is an example of a geometric sequence
2. Lizzy's marathon training schedule is an example of an arithmetic sequence
3. Lizzy will be better prepared for the marathon
Step-by-step explanation:
In the arithmetic sequence there is a common difference between each two consecutive terms
In the geometric sequence there is a common ratio between each two consecutive terms
Juan's Schedule
∵ Juan's will run one mile on the first day of the week
∴ [tex]a_{1}[/tex] = 1
∵ He will double the amount he runs each day for the next
6 days
- That means he multiplies each day by 2 to find how many miles
he will run next day
∴ [tex]a_{2}[/tex] = 1 × 2 = 2 miles
∴ [tex]a_{3}[/tex] = 2 × 2 = 4 miles
∴ [tex]a_{4}[/tex] = 4 × 2 = 8 miles
∴ [tex]a_{5}[/tex] = 8 × 2 = 16 miles
∴ [tex]a_{6}[/tex] = 16 × 2 = 32 miles
∴ [tex]a_{7}[/tex] = 32 × 2 = 64 miles
That means there is a common ratio 2 between each two consecutive days
1. Juan's marathon training schedule is an example of a geometric sequence
Lizzy's Schedule
∵ Lizzy's will run 10 miles on the first day of the week
∴ [tex]a_{1}[/tex] = 10
∵ She will increase the amount she runs by 3 miles each day for
the next six days
- That means she adds each day by 3 to find how many miles
she will run next day
∴ [tex]a_{2}[/tex] = 10 + 3 = 13 miles
∴ [tex]a_{3}[/tex] = 13 + 3 = 16 miles
∴ [tex]a_{4}[/tex] = 16 + 3 = 19 miles
∴ [tex]a_{5}[/tex] = 19 + 3 = 22 miles
∴ [tex]a_{6}[/tex] = 22 + 3 = 25 miles
∴ [tex]a_{7}[/tex] = 25 × 3 = 28 miles
That means there is a common difference 3 between each two consecutive days
2. Lizzy's marathon training schedule is an example of an arithmetic sequence
The rule of the sum of nth term in the geometric sequence is [tex]S_{n}=\frac{a_{1}(1-r^{n})}{1-r}[/tex]
∵ [tex]a_{1}[/tex] = 1 , r = 2 and n = 7
∴ [tex]S_{7}=\frac{1(1-2^{7})}{1-2}[/tex]
∴ [tex]S_{7}[/tex] = 127
∴ Juan will run 127 miles in the final week
The rule of the sum of nth term in the arithmetic sequence is [tex]S_{n}=\frac{n}{2}[a_{1}+a_{n}][/tex]
∵ n = 7, [tex]a_{1}[/tex] = 10 and [tex]a_{7}[/tex] = 28
∴ [tex]S_{7}=\frac{7}{2}(10+28)[/tex]
∴ [tex]S_{7}[/tex] = 133
∴ Lizzy will run 133 miles in the final week
∵ 133 miles > 127 miles
∴ Lizzy will run more miles than Juan
3. Lizzy will be better prepared for the marathon
