To determine after how many sessions Ann will have swum a total of 400 lengths, we need to calculate the total number of lengths swum by summing up the lengths swum in each session until she reaches a total of 400 lengths.

Steps to Solve:

1. Determine the number of sessions needed to reach the maximum length of 50 lengths per session:

Ann starts with 10 lengths and increases by 2 lengths per session. She will swim 50 lengths in the $n$-th session, which we previously calculated to be in the 21st session.

2. Calculate the total number of lengths swum in each session up to the 21st session:

This forms an arithmetic series with:
- First term ($a_1$) = 10
- Common difference ($d$) = 2
- Number of terms ($n$) = 21

The total number of lengths swum up to the 21st session is given by the sum of the first 21 terms of the arithmetic sequence:

$$S_{21}=\frac{n}{2}\times (a_1+a_n)$$

where $a_n$ is the 21st term, which is 50 lengths:

$$S_{21}=\frac{21}{2}\times (10+50)$$
$$S_{21}=\frac{21}{2}\times 60$$
$$S_{21}=21\times 30=630$$

Since the total length swum exceeds 400 lengths, Ann will reach her goal before the 21st session.

3. Find the exact session number at which she reaches a total of 400 lengths:

- Calculate the cumulative sum after each session until reaching 400 lengths.
- For sessions up to the 20th (since we need to calculate exactly):

The sum of lengths swum in the first 20 sessions is:

$$S_{20}=\frac{20}{2}\times (10+a_{20})$$
$$a_{20}=10+(20-1)\times 2=10+38=48$$
$$S_{20}=10\times (10+48)=10\times 58=580$$

- For the 21st session, Ann swims 50 lengths. If she swam a total of 580 lengths after 20 sessions, and needs only 400 lengths:

Total lengths after 19 sessions:

$$S_{19}=\frac{19}{2}\times (10+a_{19})$$
$$a_{19}=10+(19-1)\times 2=10+36=46$$
$$S_{19}=\frac{19}{2}\times 56=19\times 28=532$$

Therefore, to reach exactly 400 lengths, compute the remaining lengths needed after the 19th session:

$$400-532+50=400-532=400-532+50=50-32=18\text{ lengths}$$

Therefore, Ann would need to complete additional sessions.


Ann reaches 400 lengths in 16 sessions because she accumulates the exact 400 lengths after completing 16 sessions.

The correct answer is C. 16.