If an operator `oplus` is defined as
`4 oplus 3=4+ 5+ 6`
`5 oplus 4=5 +6+7+8`
`6 oplus 4=6 +7+8+9`
Then, what is the value of `n oplus 8` ?

Options

A. `8n+36`
B. `n+36`
C. `n+28`
D. `8n+28`

Answer

To solve the problem, we first need to understand the operator `oplus` as defined in the question. Let's break it down step by step.

### Step 1: Understand the Pattern
From the examples given:
- \( 4 \oplus 3 = 4 + 5 + 6 \)
- \( 5 \oplus 4 = 5 + 6 + 7 + 8 \)
- \( 6 \oplus 4 = 6 + 7 + 8 + 9 \)

We can see that the operator `oplus` takes a number \( n \) and an integer \( m \) and adds \( n \) to the next \( m \) consecutive integers.

### Step 2: Generalize the Expression
From the examples, we can generalize the expression for \( n \oplus m \) as:
\[ n \oplus m = n + (n + 1) + (n + 2) + \ldots + (n + m) \]
This means we are summing \( n \) and the next \( m \) integers.

### Step 3: Calculate \( n \oplus 8 \)
Now, we need to calculate \( n \oplus 8 \):
\[ n \oplus 8 = n + (n + 1) + (n + 2) + (n + 3) + (n + 4) + (n + 5) + (n + 6) + (n + 7) + (n + 8) \]

### Step 4: Simplify the Expression
This can be rewritten as:
\[ n \oplus 8 = n + n + 1 + n + 2 + n + 3 + n + 4 + n + 5 + n + 6 + n + 7 + n + 8 \]
This simplifies to:
\[ n \oplus 8 = 9n + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) \]

### Step 5: Calculate the Sum of the First 8 Integers
The sum of the first 8 integers can be calculated using the formula:
\[ \text{Sum} = \frac{m(m + 1)}{2} \]
where \( m = 8 \):
\[ \text{Sum} = \frac{8 \times 9}{2} = 36 \]

### Step 6: Final Calculation
Now substituting back into our expression:
\[ n \oplus 8 = 9n + 36 \]

### Conclusion
Thus, the value of \( n \oplus 8 \) is:
\[ \boxed{9n + 36} \]