Question

Consider the arithmetic sequence:
\[10, 12, 14, 16, \dots\]
If
\[n\] is an integer, which of these functions generate the sequence?
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
\[a(n)=16+3n\] for
\[n\geq-2\]
A
\[a(n)=16+3n\] for
\[n\geq-2\]
(Choice B)
\[b(n)=14+2n\] for
\[n\geq-1\]
B
\[b(n)=14+2n\] for
\[n\geq-1\]
(Choice C)
\[c(n)=10+3n\] for
\[n\geq0\]
C
\[c(n)=10+3n\] for
\[n\geq0\]
(Choice D)
\[d(n)=8+2n\] for
\[n\geq1\]
D
\[d(n)=8+2n\] for
\[n\geq1\]

Answer 1

To determine which of the given functions generate the arithmetic sequence \(10, 12, 14, 16, \ldots\), we first identify the general form of the sequence. The first term of the sequence is \(10\) and the common difference is \(2\). The \(n\)-th term of an arithmetic sequence can be expressed as:

\[
a(n) = a_1 + (n-1)d
\]

where \(a_1\) is the first term and \(d\) is the common difference.

For this sequence:
- \(a_1 = 10\)
- \(d = 2\)

Thus, we can express the \(n\)-th term as:

\[
a(n) = 10 + (n-1) \cdot 2
\]
\[
= 10 + 2n - 2
\]
\[
= 2n + 8
\]

Now, we analyze each of the choices provided to see if they match the sequence:

**Choice A:**
\[
a(n)=16+3n \text{ for } n\geq-2
\]
- When \(n = -2\): \(a(-2) = 16 + 3(-2) = 16 - 6 = 10\) (This matches the first term.)
- When \(n = -1\): \(a(-1) = 16 + 3(-1) = 16 - 3 = 13\) (Not part of the sequence.)
- The sequence generated is \(10, 13, 16, 19, \ldots\), which does not match.

**Choice B:**
\[
b(n)=14+2n \text{ for } n\geq-1
\]
- When \(n = -1\): \(b(-1) = 14 + 2(-1) = 14 - 2 = 12\) (Not matching the first term which is \(10\).)
- The sequence generated is \(12, 14, 16, 18, \ldots\), which does not match.

**Choice C:**
\[
c(n)=10+3n \text{ for } n\geq0
\]
- When \(n = 0\): \(c(0) = 10 + 3(0) = 10\) (This matches the first term.)
- When \(n = 1\): \(c(1) = 10 + 3(1) = 13\) (Not part of the sequence.)
- The sequence generated is \(10, 13, 16, 19, \ldots\), which does not match.

**Choice D:**
\[
d(n)=8+2n \text{ for } n\geq1
\]
- When \(n = 1\): \(d(1) = 8 + 2(1) = 8 + 2 = 10\) (This matches the first term.)
- When \(n = 2\): \(d(2) = 8 + 2(2) = 8 + 4 = 12\) (This matches the second term.)
- When \(n = 3\): \(d(3) = 8 + 2(3) = 8 + 6 = 14\) (This matches the third term.)
- Continuing this pattern gives \(10, 12, 14, 16, \ldots\), which matches the original sequence.

**Conclusion:**
The only valid choice that generates the sequence \(10, 12, 14, 16, \ldots\) is:

**Choice D: \(d(n)=8+2n\) for \(n\geq1\)**.