Question

The first five terms of a sequence are `a_{1}=47,\ a_{2}=33,\ a_{3}=19,\ a_{4}=5,\ a_{5}=-9.`



Based on this information, create an equation that can be used to find the nth term of the sequence, `a_{n}`. Move the correct answer to each box. Not all answers will be used.

Answer 1

To find a formula for the \( n \)-th term of the sequence given the first five terms \( a_1 = 47 \), \( a_2 = 33 \), \( a_3 = 19 \), \( a_4 = 5 \), and \( a_5 = -9 \), we first observe the pattern of the sequence.

Let's look at the observed values:
- \( a_1 = 47 \)
- \( a_2 = 33 \)
- \( a_3 = 19 \)
- \( a_4 = 5 \)
- \( a_5 = -9 \)

To find a potential pattern, we calculate the differences between consecutive terms:
- \( a_2 - a_1 = 33 - 47 = -14 \)
- \( a_3 - a_2 = 19 - 33 = -14 \)
- \( a_4 - a_3 = 5 - 19 = -14 \)
- \( a_5 - a_4 = -9 - 5 = -14 \)

We see that the sequence decreases by 14 with each term. This suggests that the sequence is linear and can be described by the equation of the form:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

From our observations:
- \( a_1 = 47 \)
- \( d = -14 \)

Substituting these values into the equation gives:
\[
a_n = 47 + (n-1)(-14)
\]

Simplifying this:
\[
a_n = 47 - 14(n-1)
\]
\[
a_n = 47 - 14n + 14
\]
\[
a_n = 61 - 14n
\]

Thus, the equation that can be used to find the \( n \)-th term of the sequence is:
\[
a_n = 61 - 14n
\]

This is your final answer for the equation describing the sequence.