At an event, 26 lemons are placed in a line at intervals of 4 meters from each other, with the first lemon being 20 meters away from the starting point. A contestar is required to bring all the lemons back to the starting point, bringing one lemon at a time, what is the total distance covered by that contestant to bring all the lemons to the starting point?

To find the total distance covered by the contestant, we need to calculate the distance for each trip to and from the starting point for all 26 lemons, and then sum up these distances.

### Step 1: Determine the positions of the lemons
- The first lemon is placed 20 meters away from the starting point.
- The distance between each subsequent lemon is 4 meters.

The position of the \( n \)-th lemon can be found using the formula:
\[
\text{Position of the } n\text{-th lemon} = 20 + (n-1) \times 4 \text{ meters}
\]
where \( n \) ranges from 1 to 26.

### Step 2: Calculate the distance covered for each lemon
For each lemon, the contestant has to travel to the position of the lemon and then return to the starting point. Therefore, the total distance covered for the \( n \)-th lemon is:
\[
\text{Distance covered for the } n\text{-th lemon} = 2 \times \text{Position of the } n\text{-th lemon}
\]

### Step 3: Sum up the distances for all 26 lemons
The total distance covered by the contestant is the sum of the distances for all 26 lemons:
\[
\text{Total distance} = 2 \times \sum_{n=1}^{26} \text{Position of the } n\text{-th lemon}
\]

Using the formula for the position of the \( n \)-th lemon:
\[
\text{Position of the } n\text{-th lemon} = 20 + (n-1) \times 4 = 4n + 16
\]
So, the total distance covered is:
\[
\text{Total distance} = 2 \times \sum_{n=1}^{26} (4n + 16)
\]

### Step 4: Simplify the expression
The sum can be split into two parts:
\[
\sum_{n=1}^{26} (4n + 16) = 4 \times \sum_{n=1}^{26} n + \sum_{n=1}^{26} 16
\]

1. **Sum of the first part** (arithmetic series):
\[
\sum_{n=1}^{26} n = \frac{26 \times 27}{2} = 351
\]
So:
\[
4 \times 351 = 1404
\]

2. **Sum of the second part**:
\[
\sum_{n=1}^{26} 16 = 16 \times 26 = 416
\]

So, the total sum is:
\[
1404 + 416 = 1820
\]

Finally, multiply by 2 to get the total distance:
\[
\text{Total distance} = 2 \times 1820 = 3640 \text{ meters}
\]

### Final Answer:
The contestant will cover a total distance of **3640 meters** to bring all the lemons back to the starting point.

Would you like more details or have any questions? Here are some related problems you might find interesting:

1. How would the total distance change if the distance between lemons were different?
2. What if there were fewer or more lemons?
3. How does the starting position of the first lemon affect the total distance?
4. What if the contestant could carry two lemons at a time?
5. How would the distance change if the lemons were arranged in a different geometric pattern?

**Tip:** When dealing with sequences, breaking them down into arithmetic components can simplify complex sums.

Explore how to calculate the total distance covered by a contestant bringing 26 lemons placed in a line, starting 20 meters away, back to the starting point. This problem involves arithmetic sequences and distance calculations. Suitable for Grades 7-9 students and math enthusiasts.

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