A function @ is defined on positive integers as `"@"(a) = "@"(a - 1) + 1`. If the value of `"@"(1)` is 1, then `"@"(3)` equals which one of the following?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem step by step, we will use the definition of the function "@" and the given initial condition.

### Step 1: Understand the function definition
The function "@" is defined as:
\[ "@"(a) = "@"(a - 1) + 1 \]
This means that to find the value of "@" at any integer \( a \), we need to know the value of "@" at \( a - 1 \).

### Step 2: Use the initial condition
We are given that:
\[ "@"(1) = 1 \]
This will be our base case to calculate further values.

### Step 3: Calculate "@"(2)
Using the definition of the function, we can find "@"(2):
\[ "@"(2) = "@"(2 - 1) + 1 = "@"(1) + 1 \]
Substituting the value of "@"(1):
\[ "@"(2) = 1 + 1 = 2 \]

### Step 4: Calculate "@"(3)
Next, we will calculate "@"(3) using the same definition:
\[ "@"(3) = "@"(3 - 1) + 1 = "@"(2) + 1 \]
Substituting the value of "@"(2):
\[ "@"(3) = 2 + 1 = 3 \]

### Conclusion
Thus, the value of "@"(3) is:
\[ "@"(3) = 3 \]

### Final Answer
The value of "@"(3) equals **3**.

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