a) A worm always moves along a straight line. Every day, it advances 5 m and retreats 3 m. At the end of 15 days, how far from the starting point will the worm be?

b) At the end of these 15 days, how many meters in total will this worm have walked?

c) Another worm also moves along a straight line, but in a different way from the first one. On the first day, it moves 1 m forward and 1/2 m backward. On the second day, it moves 1/2 m forward and 1/3 m backward. On the third day, it moves 1/3 m forward and 1/4 m backward, and so on. How many meters will it have walked after 1000 days?

d) Will this second worm ever manage to be 2 m away from the starting point?
Solution

a) Each day, the worm advances 5 meters and retreats 3 meters. Therefore, at the end of the day, the worm advances $5-3=2$ meters. After 15 days, the worm will be $15 \times 2=30$ meters from the starting point.

b) Each day, the worm travels $5+3=8$ meters. Therefore, after these 15 days, the worm will have traveled $8 \times 15=120$ meters.

c) This worm moves as follows:

![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag-17.jpg?height=394&width=992&top_left_y=1472&top_left_x=605)

Therefore, after 1000 days, the worm will be at $1-\frac{1}{1001}=\frac{1000}{1001}$ meters from the initial position.

d) Since on the $n$-th day the worm is at a distance of $1-\frac{1}{n+1}$ meters from the initial position, it will never be 2 meters from the initial position, because for any natural number $n$, we have:

$$
1-\frac{1}{n+1}<1<2
$$