Problem 1

Explanation

We are given the first two terms of the sequence, $a_1 = -6$ and $a_2 = -8$, and the recursive equation:

Our goal is to find the value of $a_3 + 2a_4$. We will find the values of $a_3$ and $2a_4$ step by step.

Finding the Third Term

We use the recursive equation by substituting $n = 1$:

Substitute the known values of $a_1$ and $a_2$:

Move $-18$ to the left side:

Finding Two Times the Fourth Term

Next, we use the recursive equation again with $n = 2$ to introduce the term $a_4$:

Substitute the values $a_3 = 5$ and $a_2 = -8$:

We can directly solve for $2a_4$ (since the question asks for $a_3 + 2a_4$):

Final Result

Now we sum the obtained values:

Thus, the value of $a_3 + 2a_4$ is 34.