Question

Which recursive sequence would produce the sequence 6, comma, minus, 8, comma, 20, comma, dot dot dot6,−8,20,...?
Answer
Multiple Choice Answers
a, start subscript, 1, end subscript, equals, 6, and , a, start subscript, n, end subscript, equals, minus, 2, a, start subscript, n, minus, 1, end subscript, minus, 1a
1
​
=6 and a
n
​
=−2a
n−1
​
−1
a, start subscript, 1, end subscript, equals, 6, and , a, start subscript, n, end subscript, equals, minus, 2, a, start subscript, n, minus, 1, end subscript, plus, 4a
1
​
=6 and a
n
​
=−2a
n−1
​
+4
a, start subscript, 1, end subscript, equals, 6, and , a, start subscript, n, end subscript, equals, 4, a, start subscript, n, minus, 1, end subscript, minus, 2a
1
​
=6 and a
n
​
=4a
n−1
​
−2
a, start subscript, 1, end subscript, equals, 6, and , a, start subscript, n, end subscript, equals, minus, a, start subscript, n, minus, 1, end subscript, minus, 2a
1
​
=6 and a
n
​
=−a
n−1
​
−2

Answer 1

To find the recursive formula that produces the sequence \(6, -8, 20, \ldots\), we can analyze the pattern between the terms.

1. **Identify the sequence:**
- The first term \(a_1 = 6\)
- The second term \(a_2 = -8\)
- The third term \(a_3 = 20\)

2. **Calculate the differences:**
- The difference from \(6\) to \(-8\) is \(-14\), so:
\[
a_2 - a_1 = -8 - 6 = -14
\]
- The difference from \(-8\) to \(20\) is \(28\), so:
\[
a_3 - a_2 = 20 - (-8) = 28
\]

3. **Find the recursive relationship:**
- The relationship appears to involve multiplying the previous term by \(-2\) and then correcting by adding or subtracting a constant.
- Let’s find a suitable formula of the form \(a_n = k \cdot a_{n-1} + b\) for some constants \(k\) and \(b\).

4. **From the calculations:**
- From \(a_2 = -2 \cdot a_1 + b\):
\[
-8 = -2 \cdot 6 + b \implies -8 = -12 + b \implies b = 4
\]
- From \(a_3 = -2 \cdot a_2 + b\):
\[
20 = -2 \cdot (-8) + 4 \implies 20 = 16 + 4 \implies 20 = 20
\]

This confirms that the recursive relation is consistent with our original sequence.

5. **Conclusion:**
The recursive formula is:
\[
a_1 = 6 \quad \text{and} \quad a_n = -2a_{n-1} + 4
\]

This corresponds to option **b** in your multiple choice answers:
\[
a_1 = 6 \quad \text{and} \quad a_n = -2a_{n-1} + 4
\]