# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-7/19

Exercises: Try Out - Set 7: Real exam simulation to sharpen your skills and build confidence. - Problem 19

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## Exercise 19

### Question

export const metadata = {
  title: "Problem 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given <InlineMath math="a_n" /> is the n-th term of a sequence of numbers. If <InlineMath math="a_n = a_{n+1} + 3a_{n+3}" />, <InlineMath math="a_1 = -1" />, and <InlineMath math="a_2 = 2" />, then the value of <InlineMath math="a_4 + a_1" /> is


### Choices

- [x] $$-2$$
- [ ] $$-1$$
- [ ] $$0$$
- [ ] $$1$$
- [ ] $$2$$

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

We are given a recursive formula for a sequence of numbers:

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

We also know that <InlineMath math="a_1 = -1" /> and <InlineMath math="a_2 = 2" />. We are asked to find the value of <InlineMath math="a_4 + a_1" />.

#### Determining the Value of the Fourth Term

To find <InlineMath math="a_4" />, we can use the given equation by substituting <InlineMath math="n = 1" />:

<MathContainer>
  <BlockMath math="\begin{aligned} a_1 &= a_{1+1} + 3a_{1+3} \\ a_1 &= a_2 + 3a_4 \end{aligned}" />
</MathContainer>

Next, we substitute the values <InlineMath math="a_1 = -1" /> and <InlineMath math="a_2 = 2" /> into the equation:

<MathContainer>
  <BlockMath math="\begin{aligned} -1 &= 2 + 3a_4 \\ 3a_4 &= -1 - 2 \\ 3a_4 &= -3 \\ a_4 &= \frac{-3}{3} \\ a_4 &= -1 \end{aligned}" />
</MathContainer>

#### Calculating the Sum

Now that we have found <InlineMath math="a_4 = -1" />, we can calculate <InlineMath math="a_4 + a_1" />:

<MathContainer>
  <BlockMath math="\begin{aligned} a_4 + a_1 &= (-1) + (-1) \\ &= -2 \end{aligned}" />
</MathContainer>

So, the value of <InlineMath math="a_4 + a_1" /> is <InlineMath math="-2" />.


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