Question

4) The explicit formula of a sequence is
\[f(n)=-6+2(n-1)\].
What is the first term of the sequence?
What is the common difference?

Answer 1

To analyze the given explicit formula of the sequence, we have:

\[
f(n) = -6 + 2(n - 1)
\]

### Finding the first term:

To find the first term of the sequence, we need to evaluate \(f(1)\):

\[
f(1) = -6 + 2(1 - 1) = -6 + 2 \cdot 0 = -6
\]

So, the **first term** of the sequence is \(-6\).

### Finding the common difference:

The explicit formula can be rewritten to easily see the terms of the sequence. Distributing the \(2\), we get:

\[
f(n) = -6 + 2n - 2 = 2n - 8
\]

This formula, \(f(n) = 2n - 8\), indicates that the sequence is a linear function of \(n\) with a slope of \(2\). The common difference \(d\) in an arithmetic sequence is simply the coefficient of \(n\) when the formula is expressed in the form of \(an + b\).

Thus, the **common difference** of the sequence is \(2\).

### Summary:
- The first term of the sequence is \(-6\).
- The common difference is \(2\).