Suppose that a sequence is defined as follows.
\[
a_{1}=-2, \quad a_{n}=4 a_{n-1}+5 \text { for } n \geq 2
\]
List the first four terms of the sequence.

Answer

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Answer

\(\boxed{\text{The first four terms of the sequence are -2, -3, -7, and -23}}\)

Steps

Step 1 ：Given the sequence definition: \(a_{1}=-2\) and \(a_{n}=4 a_{n-1}+5\) for \(n \geq 2\)

Step 2 ：Find the first four terms of the sequence.

Step 3 ：Calculate the second term: \(a_2 = 4a_1 + 5 = 4(-2) + 5 = -3\)

Step 4 ：Calculate the third term: \(a_3 = 4a_2 + 5 = 4(-3) + 5 = -7\)

Step 5 ：Calculate the fourth term: \(a_4 = 4a_3 + 5 = 4(-7) + 5 = -23\)

Step 6 ：\(\boxed{\text{The first four terms of the sequence are -2, -3, -7, and -23}}\)