Question

What is a formula for the nth term of the given sequence?
minus, 10, comma, minus, 19, comma, minus, 28, point, point, .
−10,−19,−28

Answer 1

To find the formula for the \( n \)th term of the given sequence \(-10, -19, -28, \ldots\), we can start by identifying the pattern in the sequence.

Let's first look at the differences between consecutive terms:

1. \(-19 - (-10) = -19 + 10 = -9\)
2. \(-28 - (-19) = -28 + 19 = -9\)

The difference between terms is constant at \(-9\). This indicates that the sequence is arithmetic, where the common difference \( d = -9 \).

An arithmetic sequence can be described with the general formula:

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

Where:
- \( a_n \) is the \( n \)th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

For this sequence:
- The first term \( a_1 = -10 \),
- The common difference \( d = -9 \).

Plugging in these values into the formula gives us:

\[
a_n = -10 + (n - 1)(-9)
\]
\[
a_n = -10 - 9(n - 1)
\]
\[
a_n = -10 - 9n + 9
\]
\[
a_n = -9n - 1
\]

Thus, the formula for the \( n \)th term of the sequence is:

\[
\boxed{-9n - 1}
\]