
the pool manager noticed that he had to add water to the pool each day because of evaporation. On the first day of summer, he added 50 gallons of water to the pool. He added 52 gallons the second day and 54 gallons the 3rd day. This trend continues. After 45 days, how much water did he add to the pool, in total?


To solve this problem, we recognize that the amount of water added to the pool each day forms an arithmetic sequence.

### Given:
- **First term** (\(a_1\)): 50 gallons on the first day.
- **Common difference** (\(d\)): 52 - 50 = 2 gallons increase each day.
- **Number of terms** (\(n\)): 45 days.

### Step 1: Identify the Arithmetic Sequence

The sequence of the amount of water added each day is:

\[
50, 52, 54, \ldots
\]

This is an arithmetic sequence with a common difference \(d = 2\).

### Step 2: Find the Amount of Water Added on the 45th Day

To find the amount of water added on the 45th day (\(a_{45}\)), we use the formula for the \(n\)-th term of an arithmetic sequence:

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

Substitute \(n = 45\), \(a_1 = 50\), and \(d = 2\):

\[
a_{45} = 50 + (45 - 1) \cdot 2
\]

\[
a_{45} = 50 + 44 \cdot 2
\]

\[
a_{45} = 50 + 88 = 138
\]

So, the pool manager added **138 gallons** of water on the 45th day.

### Step 3: Find the Total Amount of Water Added Over 45 Days

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

Substitute \(n = 45\), \(a_1 = 50\), and \(a_{45} = 138\):

\[
S_{45} = \frac{45}{2} \cdot (50 + 138)
\]

\[
S_{45} = \frac{45}{2} \cdot 188
\]

\[
S_{45} = 45 \cdot 94
\]

\[
S_{45} = 4230
\]

### Final Answer

The pool manager added a total of **4,230 gallons** of water over the 45 days.

---

Would you like more details or have any questions? Here are some related questions to consider:

1. What would be the total amount added if the trend continued for 60 days?
2. How would the total change if the pool manager started with 60 gallons on the first day?
3. How many gallons of water were added by the 30th day?
4. What is the average number of gallons added per day over the 45 days?
5. How does the arithmetic sequence formula help in solving real-life problems like this?

**Tip:** Understanding the pattern in arithmetic sequences can help solve problems involving consistent changes, like daily increments or decrements.

The pool manager noticed that he had to add water to the pool each day because of evaporation. On the first day of summer, he added 50 gallons of water to the pool. He added 52 gallons the second day and 54 gallons the 3rd day. This trend continues. After 45 days, how much water did he add to the pool, in total?

Discover how to calculate the total amount of water added to a pool over 45 days using arithmetic sequences. This problem involves an increasing daily trend and uses key formulas to determine the final sum. Ideal for students in Grades 9-10.

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