# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-8/4

Exercises: Try Out - Set 8: Real exam simulation to sharpen your skills and build confidence. - Problem 4

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## Exercise 4

### Question

export const metadata = {
  title: "Problem 4",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

<InlineMath math="x_0" /> is the average of data <InlineMath math="x_1, x_2, ..., x_{10}" />. If the data changes following the pattern <InlineMath math="\frac{x_1}{2} + 3, \frac{x_2}{2} + 5, \frac{x_3}{2} + 7" />, and so on, then the average value becomes...


### Choices

- [ ] $$x_0 + 11$$
- [ ] $$x_0 + 12$$
- [ ] $$\frac{1}{2}x_0 + 11$$
- [x] $$\frac{1}{2}x_0 + 12$$
- [ ] $$\frac{1}{2}x_0 + 21$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 4",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

The new data follows the pattern:

<MathContainer>
  <BlockMath math="\frac{x_1}{2} + 3, \frac{x_2}{2} + 5, \frac{x_3}{2} + 7, ..." />
</MathContainer>

The new average can be obtained by:

<MathContainer>
  <BlockMath math="\bar{x} = \frac{\left(\frac{x_1}{2} + 3\right) + \left(\frac{x_2}{2} + 5\right) + \left(\frac{x_3}{2} + 7\right) + ... + \left(\frac{x_{10}}{2} + (2(10)+1)\right)}{10}" />
</MathContainer>

First, let's find the sum of the arithmetic series <InlineMath math="3, 5, 7, ..." />. This sequence has the first term <InlineMath math="a = 3" /> and difference <InlineMath math="b = 2" />. The 10th term is:

<MathContainer>
  <BlockMath math="U_{10} = 3 + 9(2) = 21" />
</MathContainer>

The sum of the first 10 terms is:

<MathContainer>
  <BlockMath math="S_{10} = \frac{10}{2}(3 + 21) = 5(24) = 120" />
</MathContainer>

The new average can be written as:

<MathContainer>
  <BlockMath math="\begin{aligned} \bar{x} &= \frac{\frac{1}{2}(x_1 + x_2 + ... + x_{10}) + 120}{10} \\ &= \frac{1}{2}\frac{(x_1 + x_2 + ... + x_{10})}{10} + \frac{120}{10} \\ &= \frac{1}{2}x_0 + 12 \end{aligned}" />
</MathContainer>

Thus, the new average is <InlineMath math="\frac{1}{2}x_0 + 12" />.


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