If the arithmetic mean of 5 consecutive numbers is 16, find the sum of the smallest and the largest integer.

Answer

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

### Step 1: Understand the Problem
We need to find the sum of the smallest and largest integers among 5 consecutive numbers whose arithmetic mean is 16.

### Step 2: Set Up the Consecutive Numbers
Let’s denote the 5 consecutive numbers. If we assume the middle number is 16 (since the mean is 16), the numbers can be represented as:
- Smallest number: \( x \)
- Next number: \( x + 1 \)
- Middle number: \( x + 2 \) (which is 16)
- Next number: \( x + 3 \)
- Largest number: \( x + 4 \)

### Step 3: Find the Value of \( x \)
Since the middle number is 16:
\[ x + 2 = 16 \]
To find \( x \), we can rearrange the equation:
\[ x = 16 - 2 \]
\[ x = 14 \]

### Step 4: List the Consecutive Numbers
Now that we have \( x = 14 \), we can list the 5 consecutive numbers:
- Smallest: \( 14 \)
- Next: \( 15 \)
- Middle: \( 16 \)
- Next: \( 17 \)
- Largest: \( 18 \)

### Step 5: Find the Sum of the Smallest and Largest Numbers
Now we need to find the sum of the smallest and largest integers:
\[ \text{Sum} = 14 + 18 \]

### Step 6: Calculate the Sum
Calculating the sum:
\[ 14 + 18 = 32 \]

### Final Answer
The sum of the smallest and largest integers is **32**.

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