Tobi is training to run a marathon. In the first week she runs 25 km and increases this distance by 10% each week. This situation may be modelled by the series 25 + 25(1.1) + 25(1.1)2 + . She wishes to continue this pattern for 14 weeks. How far will she have run in total when she completes the 14th week? Express your answer to the nearest tenth of a kilometre.
Kane is training for a marathon. He starts by running 3 miles during every training session each week plans to increase the distance of his runs by 1/4 mile. Let w be the number of weeks. Write and expression to show the diatance kane runs in a training session after w weeks
To find out how far Tobi will have run in total after completing the 14th week, we need to calculate the sum of the terms in the series.
The series can be written as 25 + 25(1.1) + 25(1.1)^2 + ...
To find the sum of the terms in this geometric series, we can use the formula for the sum of a geometric series:
Sum = a(1 - r^n) / (1 - r)
where:
- Sum is the total sum of the series,
- "a" is the first term in the series,
- "r" is the common ratio,
- "n" is the number of terms in the series.
In this case, the first term (a) is 25, the common ratio (r) is 1.1, and the number of terms (n) is 14.
Substituting these values into the formula, we get:
Sum = 25(1 - 1.1^14) / (1 - 1.1)
Calculating this, we find that Tobi will have run approximately 171.2 kilometers when she completes the 14th week.
To find out how far Tobi will have run in the 14th week, we need to find the sum of the geometric series.
The formula to find the sum of a geometric series is given by:
S = a(1 - r^n) / (1 - r),
where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
In this case, Tobi runs 25 km in the first week, and each subsequent week she increases the distance by 10%, which means the common ratio is 1.1 (since she is multiplying the previous week's distance by 1.1).
Using the formula, we plug in the values:
a = 25 (first term)
r = 1.1 (common ratio)
n = 14 (number of terms)
S = 25(1 - 1.1^14) / (1 - 1.1).
To simplify, we need to calculate 1.1^14 to get:
S = 25(1 - 2.853116706) / (-0.1)
S = -25 * 1.853116706 / (-0.1)
S = 46.32791765 / 0.1
S = 463.2791765 km.
Therefore, when Tobi completes the 14th week, she will have run approximately 463.3 km in total.
just use what you know about geometric sequences:
S14 = 25(1.1^14-1)/(1.14-1) = 499.55 km