To determine after how many sessions Ann will swim 50 lengths for the first time, we can model the problem as an arithmetic sequence where the number of lengths Ann swims each session increases by 2 from the previous session.

Given:
- The number of lengths Ann swims in the first session (\(a_1\)) is 10.
- The number of lengths increases by 2 each session (common difference \(d = 2\)).
- Ann will swim 50 lengths for the first time when the number of lengths reaches 50.

We need to find the session number (\(n\)) where Ann swims 50 lengths.

Step-by-Step Solution:

1. Use the formula for the \(n\)-th term of an arithmetic sequence:

\[
a_n = a_1 + (n - 1) \times d
\]

Here, \(a_n = 50\), \(a_1 = 10\), and \(d = 2\). Substitute these values into the formula:

\[
50 = 10 + (n - 1) \times 2
\]

2. Solve for \(n\):

\[
50 = 10 + 2(n - 1)
\]
\[
50 = 10 + 2n - 2
\]
\[
50 = 8 + 2n
\]
\[
50 - 8 = 2n
\]
\[
42 = 2n
\]
\[
n = \frac{42}{2} = 21
\]


Ann will swim 50 lengths for the first time after 21 sessions.

The correct answer is A. 21.