# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/1

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 1

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## Exercise 1

### Question

export const metadata = {
  title: "Problem 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given <InlineMath math="a_1 = -6" /> and <InlineMath math="a_2 = -8" />. If the recurrence relation satisfies the equation <InlineMath math="a_{n+1} = 2a_{n+2} + 3a_n" />, then the value of <InlineMath math="a_3 + 2a_4" /> is ...


### Choices

- [ ] $$39$$
- [ ] $$29$$
- [x] $$34$$
- [ ] $$24$$
- [ ] $$16$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are given the first two terms of the sequence, <InlineMath math="a_1 = -6" /> and <InlineMath math="a_2 = -8" />, and the recursive equation:

<MathContainer>
  <BlockMath math="a_{n+1} = 2a_{n+2} + 3a_n" />
</MathContainer>

Our goal is to find the value of <InlineMath math="a_3 + 2a_4" />. We will find the values of <InlineMath math="a_3" /> and <InlineMath math="2a_4" /> step by step.

#### Finding the Third Term

We use the recursive equation by substituting <InlineMath math="n = 1" />:

<MathContainer>
  <BlockMath math="a_{1+1} = 2a_{1+2} + 3a_1" />
  <BlockMath math="a_2 = 2a_3 + 3a_1" />
</MathContainer>

Substitute the known values of <InlineMath math="a_1" /> and <InlineMath math="a_2" />:

<MathContainer>
  <BlockMath math="-8 = 2a_3 + 3(-6)" />
  <BlockMath math="-8 = 2a_3 - 18" />
</MathContainer>

Move <InlineMath math="-18" /> to the left side:

<MathContainer>
  <BlockMath math="2a_3 = -8 + 18" />
  <BlockMath math="2a_3 = 10" />
  <BlockMath math="a_3 = 5" />
</MathContainer>

#### Finding Two Times the Fourth Term

Next, we use the recursive equation again with <InlineMath math="n = 2" /> to introduce the term <InlineMath math="a_4" />:

<MathContainer>
  <BlockMath math="a_{2+1} = 2a_{2+2} + 3a_2" />
  <BlockMath math="a_3 = 2a_4 + 3a_2" />
</MathContainer>

Substitute the values <InlineMath math="a_3 = 5" /> and <InlineMath math="a_2 = -8" />:

<MathContainer>
  <BlockMath math="5 = 2a_4 + 3(-8)" />
  <BlockMath math="5 = 2a_4 - 24" />
</MathContainer>

We can directly solve for <InlineMath math="2a_4" /> (since the question asks for <InlineMath math="a_3 + 2a_4" />):

<MathContainer>
  <BlockMath math="2a_4 = 5 + 24" />
  <BlockMath math="2a_4 = 29" />
</MathContainer>

#### Final Result

Now we sum the obtained values:

<MathContainer>
  <BlockMath math="a_3 + 2a_4 = 5 + 29 = 34" />
</MathContainer>

Thus, the value of <InlineMath math="a_3 + 2a_4" /> is 34.


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