Question

Juan and Lizzy are in the final week of their training for a marathon. Juan's goal is to run one mile on the first day of the week and double the amount he runs each day for the next six days. Lizzy's goal is to run 10 miles on the first day of the week and increase the amount she runs by 3 miles each day for the next six days. 1: Juan's marathon training schedule is an example of a(n) : arithmetic or geometric 2: Lizzy's marathon training schedule is an example of a(n) : arithmetic or geometric 3: Who will be better prepared for the marathon: Juan or Lizzy

Answer 1

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

∴ $a_{1}$ = 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

∴  $a_{2}$ = 1 × 2 = 2 miles

∴  $a_{3}$ = 2 × 2 = 4 miles

∴  $a_{4}$ = 4 × 2 = 8 miles

∴  $a_{5}$ = 8 × 2 = 16 miles

∴  $a_{6}$ = 16 × 2 = 32 miles

∴  $a_{7}$ = 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

∴ $a_{1}$ = 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

∴  $a_{2}$ = 10 + 3 = 13 miles

∴  $a_{3}$ = 13 + 3 = 16 miles

∴  $a_{4}$ = 16 + 3 = 19 miles

∴  $a_{5}$ = 19 + 3 = 22 miles

∴  $a_{6}$ = 22 + 3 = 25 miles

∴  $a_{7}$ = 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 $S_{n}=\frac{a_{1}(1-r^{n})}{1-r}$

∵ $a_{1}$ = 1 , r = 2 and n = 7

∴  $S_{7}=\frac{1(1-2^{7})}{1-2}$

∴  $S_{7}$ = 127

∴ Juan will run 127 miles in the final week

The rule of the sum of nth term in the arithmetic sequence is $S_{n}=\frac{n}{2}[a_{1}+a_{n}]$

∵ n = 7,  $a_{1}$ = 10  and  $a_{7}$ = 28

∴ $S_{7}=\frac{7}{2}(10+28)$

∴ $S_{7}$ = 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