Question

What is a formula for the nth term of the given sequence?
minus, 8, comma, 1, comma, 10, dot dot dot
−8,1,10...
Answer
Multiple Choice Answers
a, start subscript, n, end subscript, equals, minus, 8, left parenthesis, 9, right parenthesis, to the power na
n
​
=−8(9)
n

a, start subscript, n, end subscript, equals, 1, plus, 9, left parenthesis, n, minus, 1, right parenthesisa
n
​
=1+9(n−1)
a, start subscript, n, end subscript, equals, minus, 26, plus, 9, left parenthesis, n, plus, 1, right parenthesisa
n
​
=−26+9(n+1)
a, start subscript, n, end subscript, equals, minus, 8, plus, 9, left parenthesis, n, plus, 1, right parenthesisa
n
​
=−8+9(n+1)

Answer 1

To find the formula for the nth term of the sequence \(-8, 1, 10, \ldots\), we first notice the changes between consecutive terms:

- From \(-8\) to \(1\) is an increase of \(9\).
- From \(1\) to \(10\) is also an increase of \(9\).

This suggests that the sequence is an arithmetic sequence, where the first term \(a_1\) is \(-8\) and the common difference \(d\) is \(9\).

The formula for the nth term of an arithmetic sequence is given by:

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

Substituting the values we have:

\[
a_n = -8 + (n - 1) \cdot 9
\]

This simplifies to:

\[
a_n = -8 + 9n - 9
\]
\[
a_n = 9n - 17
\]

Now, let's verify which of the provided options corresponds to our derived formula.

1. \( a_n = -8(9)^n \) (not a linear formula)
2. \( a_n = 1 + 9(n - 1) \) (this simplifies to \( 9n - 8 \)) - **This is close but not the match.**
3. \( a_n = -26 + 9(n + 1) \) (this simplifies to \( 9n - 17 \)) - **This is a match!**
4. \( a_n = -8 + 9(n + 1) \) (this simplifies to \( 9n + 1\)) - **Doesn’t match.**

Thus, the correct formula for the nth term of the sequence is:

\[
\boxed{a_n = -26 + 9(n + 1)}
\]