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.

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.
failed

Solution

failed
failed

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:

\[ a_4 = -2a_3 + 6 = -2(14) + 6 = -28 + 6 = -22 \]

Final Answer

The first four terms of the sequence are:

\[ \boxed{5, -4, 14, -22} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful