Questions: Suppose that a sequence is defined as follows.
a1=5, an=-2 an-1+6 for n geq 2
List the first four terms of the sequence.
Transcript text: Suppose that a sequence is defined as follows.
\[
a_{1}=5, \quad a_{n}=-2 a_{n-1}+6 \text { for } n \geq 2
\]
List the first four terms of the sequence.
Solution
Solution Steps
Step 1: Determine the First Term
The first term of the sequence is given as \( a_1 = 5 \).
Step 2: Calculate the Second Term
Using the recursive formula \( a_n = -2a_{n-1} + 6 \), we can find the second term:
\[
a_2 = -2a_1 + 6 = -2(5) + 6 = -10 + 6 = -4
\]
Step 3: Calculate the Third Term
Using the same recursive formula, we find the third term:
\[
a_3 = -2a_2 + 6 = -2(-4) + 6 = 8 + 6 = 14
\]
Step 4: Calculate the Fourth Term
Finally, using the recursive formula again, we find the fourth term: