Question: A machine follows $M(n) = 8n - 3$ for $n = 1, 2, 3, \ldots$. A student says the constant term $-3$ means the first output is $-3$. Is this correct?
1. **State the problem:** We have a machine defined by the function $$M(n) = 8n - 3$$ where $n$ is a positive integer starting from 1. The question is whether the constant term $-3$ means the first output (when $n=1$) is $-3$.
2. **Understand the formula:** The function $M(n) = 8n - 3$ means for each input $n$, the output is calculated by multiplying $n$ by 8 and then subtracting 3.
3. **Evaluate the first output:** Substitute $n=1$ into the formula:
$$M(1) = 8 \times 1 - 3 = 8 - 3 = 5$$
4. **Interpretation:** The constant term $-3$ is part of the formula and affects all outputs by subtracting 3, but it does not mean the first output is $-3$. Instead, the first output is $5$.
5. **Conclusion:** The student's statement is incorrect. The constant term $-3$ does not represent the first output; it is just part of the formula that adjusts the output values.