If the mean of 5 consecutive odd integers is 13, find the sum of the largest and the smallest integers.

Answer

To solve the problem step by step, follow these instructions:

### Step 1: Understand the Problem
We need to find five consecutive odd integers whose mean is 13, and then find the sum of the largest and smallest integers among them.

### Step 2: Define the Odd Integers
Let the smallest odd integer be represented as \( a \). The next four consecutive odd integers can be expressed as:
- First integer: \( a \)
- Second integer: \( a + 2 \)
- Third integer: \( a + 4 \)
- Fourth integer: \( a + 6 \)
- Fifth integer: \( a + 8 \)

### Step 3: Set Up the Mean Equation
The mean (average) of these five integers is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
In our case, the mean is 13, and we have 5 observations. Therefore, we can write:
\[ 13 = \frac{a + (a + 2) + (a + 4) + (a + 6) + (a + 8)}{5} \]

### Step 4: Simplify the Equation
Combine the terms in the numerator:
\[ 13 = \frac{5a + (2 + 4 + 6 + 8)}{5} \]
Calculating the sum of the constants:
\[ 2 + 4 + 6 + 8 = 20 \]
So the equation becomes:
\[ 13 = \frac{5a + 20}{5} \]

### Step 5: Eliminate the Denominator
Multiply both sides by 5 to eliminate the fraction:
\[ 65 = 5a + 20 \]

### Step 6: Solve for \( a \)
Now, isolate \( 5a \) by subtracting 20 from both sides:
\[ 5a = 65 - 20 \]
\[ 5a = 45 \]
Now, divide by 5:
\[ a = \frac{45}{5} = 9 \]

### Step 7: Find the Five Odd Integers
Now that we have \( a = 9 \), we can find the five consecutive odd integers:
- First: \( 9 \)
- Second: \( 11 \) (which is \( 9 + 2 \))
- Third: \( 13 \) (which is \( 9 + 4 \))
- Fourth: \( 15 \) (which is \( 9 + 6 \))
- Fifth: \( 17 \) (which is \( 9 + 8 \))

### Step 8: Identify the Largest and Smallest Integers
The smallest integer is \( 9 \) and the largest integer is \( 17 \).

### Step 9: Calculate the Sum
Now, we need to find the sum of the largest and smallest integers:
\[ \text{Sum} = 9 + 17 = 26 \]

### Final Answer
The sum of the largest and smallest integers is \( 26 \).

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