Question

The first five terms of a sequence are "a" Subscript, 1 , Baseline equals 47 , "a" Subscript, 2 , Baseline equals 33 , "a" Subscript, 3 , Baseline equals 19 , "a" Subscript, 4 , Baseline equals 5 , "a" Subscript, 5 , Baseline equals negative 9.a1​=47, a2​=33, a3​=19, a4​=5, a5​=−9.

Based on this information, create an equation that can be used to find the nth term of the sequence, "a" Subscript, "n" , Baselinean​.

Answer 1

To find a general formula for the \( n \)-th term of the sequence given:

- \( a_1 = 47 \)
- \( a_2 = 33 \)
- \( a_3 = 19 \)
- \( a_4 = 5 \)
- \( a_5 = -9 \)

we first observe the differences between consecutive terms.

1. \( a_2 - a_1 = 33 - 47 = -14 \)
2. \( a_3 - a_2 = 19 - 33 = -14 \)
3. \( a_4 - a_3 = 5 - 19 = -14 \)
4. \( a_5 - a_4 = -9 - 5 = -14 \)

The sequence has a constant difference of \(-14\), indicating that it is an arithmetic sequence.

The general form for the \( n \)-th term of an arithmetic sequence can be expressed as:

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

where:
- \( a_1 \) is the first term of the sequence,
- \( d \) is the common difference,
- \( n \) is the term number.

Substituting the values we have:
- \( a_1 = 47 \)
- \( d = -14 \)

Thus, the formula becomes:

\[
a_n = 47 + (n - 1)(-14)
\]

We can simplify this expression:

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

Therefore, the equation to find the \( n \)-th term of the sequence is:

\[
a_n = 61 - 14n
\]