# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-7/5

Exercises: Try Out - Set 7: Real exam simulation to sharpen your skills and build confidence. - Question 5

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## Exercise 5

### Question

export const metadata = {
  title: "Question 5",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given that <InlineMath math="a_n" /> is the <InlineMath math="n" />-th term of a sequence. If <InlineMath math="a_{n+1} = a_{n+2} + 2a_n" />, <InlineMath math="a_3 = -1" />, and <InlineMath math="a_5 = 3" />, then the value of <InlineMath math="a_2" /> is ...


### Choices

- [ ] $$-3$$
- [x] $$-1$$
- [ ] $$0$$
- [ ] $$1$$
- [ ] $$3$$

### Answer & Explanation

export const metadata = {
  title: "Question 5 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the recursive formula for the sequence:

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

We are given <InlineMath math="a_3 = -1" /> and <InlineMath math="a_5 = 3" />. We need to find the value of <InlineMath math="a_2" />.

First, we find the value of <InlineMath math="a_4" /> by using the recursive formula and substituting <InlineMath math="n = 3" /> to relate <InlineMath math="a_3" />, <InlineMath math="a_4" />, and <InlineMath math="a_5" />.

<MathContainer>
  <BlockMath math="\begin{align*} a_{3+1} &= a_{3+2} + 2a_3 \\ a_4 &= a_5 + 2a_3 \end{align*}" />
</MathContainer>

Substitute the values <InlineMath math="a_5 = 3" /> and <InlineMath math="a_3 = -1" />:

<MathContainer>
  <BlockMath math="\begin{align*} a_4 &= 3 + 2(-1) \\ a_4 &= 3 - 2 \\ a_4 &= 1 \end{align*}" />
</MathContainer>

Next, we find the value of <InlineMath math="a_2" />. Since we now have the values of <InlineMath math="a_3" /> and <InlineMath math="a_4" />, we can use the recursive formula with <InlineMath math="n = 2" /> to relate <InlineMath math="a_2" />, <InlineMath math="a_3" />, and <InlineMath math="a_4" />.

<MathContainer>
  <BlockMath math="\begin{align*} a_{2+1} &= a_{2+2} + 2a_2 \\ a_3 &= a_4 + 2a_2 \end{align*}" />
</MathContainer>

Substitute the values <InlineMath math="a_3 = -1" /> and <InlineMath math="a_4 = 1" />:

<MathContainer>
  <BlockMath math="\begin{align*} -1 &= 1 + 2a_2 \\ -1 - 1 &= 2a_2 \\ -2 &= 2a_2 \\ a_2 &= -1 \end{align*}" />
</MathContainer>

Thus, the value of <InlineMath math="a_2" /> is <InlineMath math="-1" />.


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