While training for a race, a runner increases her distance by 10% each day. If she runs 2km on the first day, what will her total distance for 26 days of training?

42

2* (1 +1.1 + 1.1^2+ 1.1^3 +....1.1^25

geometric series with r = 1.1, n=26, a = 1
2 [ar^0 + ar^1 + ar^2 .. + a r^(n-1)]
=2 [ (r^n -1) / (r-1) ]
=2 [ (10.9)/(.1)]
=218km

To find the total distance for 26 days of training, we need to calculate the distance covered each day and add them up:

1. On the first day, the runner runs 2km.
2. On the second day, the distance increases by 10%, which means the runner will run 2km + 10% of 2km = 2km + 0.2km = 2.2km.
3. On the third day, the distance increases again by 10%, so the runner will run 2.2km + 10% of 2.2km = 2.2km + 0.22km = 2.42km.

We can see that on each subsequent day, the runner will add 10% of the previous day's distance to the total.

Therefore, for the 26 days of training, we need to calculate the distance for each day and add them up:

Total distance for 26 days of training = 2km + 2.2km + 2.42km + ... (and so on for 26 days)

To simplify this calculation, we can use the formula for the sum of the first n terms of an arithmetic series:

Sum = n/2 * (2a + (n-1)d)

In our case:
a = 2km (the first term)
d = 10% of 2km = 0.1 * 2km = 0.2km (the common difference)
n = 26 (the number of terms)

Plugging these values into the formula:

Total distance for 26 days of training = 26/2 * (2(2km) + (26-1)(0.2km))

Now let's solve this equation step by step.

Step 1: Calculate the term inside the parentheses.
Inside parentheses = 2(2km) + (26-1)(0.2km)
= 4km + 25(0.2km)
= 4km + 5km
= 9km

Step 2: Calculate the numerator of the fraction in the formula.
Numerator = 26/2 * (9km)
= 13 * 9km
= 117km

Step 3: Calculate the total distance for 26 days of training.
Total distance = 117km

Therefore, the runner's total distance for 26 days of training will be 117km.

To find the runner's total distance for 26 days of training, you need to calculate the distance she runs each day and sum them up.

Since the runner increases her distance by 10% each day, you can calculate the distance for each day using the formula:

distance = initial distance * (1 + increase percentage)

On the first day, the runner runs 2 km. To calculate the distance for the second day, you can use the formula:

distance for the second day = 2 km * (1 + 10%)

To calculate the distance for the third day, you can use the formula:

distance for the third day = distance for the second day * (1 + 10%)

You can repeat this process for each day until you reach the 26th day.

Here is a step-by-step breakdown of finding the runner's total distance for 26 days of training:

1. Calculate the distance for the first day:
distance for the first day = 2 km

2. Calculate the distance for the second day:
distance for the second day = distance for the first day * (1 + 10%)

3. Calculate the distance for the third day:
distance for the third day = distance for the second day * (1 + 10%)

4. Repeat steps 2 and 3 for each day until you reach the 26th day.

5. Once you have calculated the distance for each day, sum them up to find the runner's total distance for 26 days of training.

By following these steps, you can calculate the runner's total distance for 26 days of training.