Ann goes swimming regularly. She wants to improve her fitness so she decides to swim 10 lengths in the first session and increase the number of lengths she swims by 2 every session. When she reaches 50 lengths in a session she will not increase the number any further. Ann decides she will give herself a reward when she has swum a total of 400 lengths. After how many sessions does she get her reward?
A. 15 B. 25 C. 16 D. 24
Correct Answer: C
Explanation
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:
