Question
Liliana is recovering from a major medical procedure. She can’t walk much to start, but she has a goal to walk a total of 85932 steps in 10 weeks. If she doubles the amount of steps that she takes per week, how many steps does she have to take in week one in order to reach her goal? Round to the nearest step.
Answers
GPT-4o mini
To find out how many steps Liliana needs to take in week one in order to reach her goal, we first need to determine the total number of steps she must take each week.
Liliana's goal is 85,932 steps over 10 weeks. We can calculate the average number of steps she needs to take each week:
\[
\text{Average steps per week} = \frac{85,932 \, \text{steps}}{10 \, \text{weeks}} = 8,593.2 \, \text{steps per week}
\]
Since we need to account for her doubling her steps each week, let's denote the number of steps she takes in week one as \( x \).
If she doubles the steps each subsequent week, then the steps she takes will be:
- Week 1: \( x \)
- Week 2: \( 2x \)
- Week 3: \( 4x \)
- Week 4: \( 8x \)
- Week 5: \( 16x \)
- Week 6: \( 32x \)
- Week 7: \( 64x \)
- Week 8: \( 128x \)
- Week 9: \( 256x \)
- Week 10: \( 512x \)
To find the total number of steps over the 10 weeks, we can sum these amounts:
\[
\text{Total steps} = x + 2x + 4x + 8x + 16x + 32x + 64x + 128x + 256x + 512x
\]
This is a geometric series where \( a = x \) (the first term) and the common ratio \( r = 2 \). The total number of terms is \( n = 10 \). The sum \( S \) of a geometric series can be calculated using the formula:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Plugging in the values:
\[
S_{10} = x \frac{2^{10} - 1}{2 - 1} = x (1024 - 1) = 1023x
\]
Now we set this equal to her goal of 85,932 steps:
\[
1023x = 85,932
\]
Now, we solve for \( x \):
\[
x = \frac{85,932}{1023}
\]
Calculating that:
\[
x \approx 84.15
\]
Rounding to the nearest step, we get:
\[
x \approx 84
\]
Thus, the number of steps Liliana has to take in week one is **84** steps.
Liliana's goal is 85,932 steps over 10 weeks. We can calculate the average number of steps she needs to take each week:
\[
\text{Average steps per week} = \frac{85,932 \, \text{steps}}{10 \, \text{weeks}} = 8,593.2 \, \text{steps per week}
\]
Since we need to account for her doubling her steps each week, let's denote the number of steps she takes in week one as \( x \).
If she doubles the steps each subsequent week, then the steps she takes will be:
- Week 1: \( x \)
- Week 2: \( 2x \)
- Week 3: \( 4x \)
- Week 4: \( 8x \)
- Week 5: \( 16x \)
- Week 6: \( 32x \)
- Week 7: \( 64x \)
- Week 8: \( 128x \)
- Week 9: \( 256x \)
- Week 10: \( 512x \)
To find the total number of steps over the 10 weeks, we can sum these amounts:
\[
\text{Total steps} = x + 2x + 4x + 8x + 16x + 32x + 64x + 128x + 256x + 512x
\]
This is a geometric series where \( a = x \) (the first term) and the common ratio \( r = 2 \). The total number of terms is \( n = 10 \). The sum \( S \) of a geometric series can be calculated using the formula:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Plugging in the values:
\[
S_{10} = x \frac{2^{10} - 1}{2 - 1} = x (1024 - 1) = 1023x
\]
Now we set this equal to her goal of 85,932 steps:
\[
1023x = 85,932
\]
Now, we solve for \( x \):
\[
x = \frac{85,932}{1023}
\]
Calculating that:
\[
x \approx 84.15
\]
Rounding to the nearest step, we get:
\[
x \approx 84
\]
Thus, the number of steps Liliana has to take in week one is **84** steps.