Question

First four terms for an arithmetic sequence are -11 -5 1 7 what is the equation for a(under)n

Answer 1

An arithmetic sequence is defined by the formula:

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

where:
- \( a(n) \) is the \( n \)-th term,
- \( a(1) \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

From the sequence you provided, the first four terms are:
- \( a(1) = -11 \)
- \( a(2) = -5 \)
- \( a(3) = 1 \)
- \( a(4) = 7 \)

To find the common difference \( d \), we can subtract the first term from the second term:

\[
d = a(2) - a(1) = -5 - (-11) = -5 + 11 = 6
\]

We can verify \( d \) by checking the difference between other consecutive terms:

\[
a(3) - a(2) = 1 - (-5) = 1 + 5 = 6
\]
\[
a(4) - a(3) = 7 - 1 = 6
\]

The common difference \( d \) is indeed 6.

Now, substituting \( a(1) = -11 \) and \( d = 6 \) into the arithmetic sequence formula:

\[
a(n) = -11 + (n - 1) \cdot 6
\]

This can be simplified:

\[
a(n) = -11 + 6n - 6
\]
\[
a(n) = 6n - 17
\]

So, the equation for the \( n \)-th term of the arithmetic sequence is:

\[
a(n) = 6n - 17
\]