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**Question 38**

38.1 The first four terms of a quadratic number pattern are [tex]-1, x, 3, x+8[/tex].

38.1.1 Calculate the value(s) of [tex]x[/tex].

38.1.2 If [tex]x=0[/tex], determine the position of the first term in the quadratic number pattern for which the sum of the first [tex]n[/tex] first differences will be greater than 250.

Answer :

Let's solve the problem step-by-step.

### 38.1.1 Calculate the value(s) of [tex]\( x \)[/tex]

We're given the first four terms of a quadratic number pattern: [tex]\(-1\)[/tex], [tex]\(x\)[/tex], [tex]\(3\)[/tex], and [tex]\(x + 8\)[/tex].

In a quadratic sequence, the second differences of the terms are constant. Let's find the first differences and then the second differences:

1. First Differences:
- Between the first and second terms: [tex]\( x - (-1) = x + 1 \)[/tex]
- Between the second and third terms: [tex]\( 3 - x \)[/tex]
- Between the third and fourth terms: [tex]\( (x + 8) - 3 = x + 5 \)[/tex]

2. Second Differences:
- Between the first and second first differences: [tex]\( (3 - x) - (x + 1) = 3 - x - x - 1 = 2 - 2x \)[/tex]
- Between the second and third first differences: [tex]\( (x + 5) - (3 - x) = x + 5 - 3 + x = 2x + 2 \)[/tex]

In a quadratic sequence, these second differences must be equal:

[tex]\[ 2 - 2x = 2x + 2 \][/tex]

3. Solve for [tex]\( x \)[/tex]:

[tex]\[
2 - 2x = 2x + 2
\][/tex]

[tex]\[
2 - 2 = 4x \quad \Rightarrow \quad 0 = 4x
\][/tex]

[tex]\[
x = 0
\][/tex]

So, the value of [tex]\( x \)[/tex] is [tex]\( 0 \)[/tex].

### 38.1.2 Determine the position of the first term for which the sum of the first [tex]\( n \)[/tex] first differences will be greater than 250, given [tex]\( x = 0 \)[/tex].

Let's find the first differences again with [tex]\( x = 0 \)[/tex]:

1. First Differences when [tex]\( x = 0 \)[/tex]:
- Between [tex]\(-1\)[/tex] and [tex]\(0\)[/tex]: [tex]\(0 - (-1) = 1\)[/tex]
- Between [tex]\(0\)[/tex] and [tex]\(3\)[/tex]: [tex]\(3 - 0 = 3\)[/tex]
- Between [tex]\(3\)[/tex] and [tex]\(8\)[/tex]: [tex]\(8 - 3 = 5\)[/tex]

Now, the pattern of first differences is [tex]\(1, 3, 5\)[/tex], which repeats every 3 terms.

2. Find [tex]\( n \)[/tex] such that the sum of first [tex]\( n \)[/tex] first differences is greater than 250:

The sequence of first differences is periodic and repeats every three terms as [tex]\(1, 3, 5\)[/tex].

Let's calculate the sum of these differences:

- Over one complete cycle (3 terms), the sum is [tex]\(1 + 3 + 5 = 9\)[/tex].

3. Finding [tex]\( n \)[/tex]:

- Find how many complete cycles give a sum exceeding 250:
- If 27 complete cycles, sum = [tex]\(9 \times 27 = 243\)[/tex]
- We require more than 27 cycles.

- Add terms from the next cycle:
- Adding the next '1' gives [tex]\(243 + 1 = 244\)[/tex]
- Adding '3' gives [tex]\(244 + 3 = 247\)[/tex]
- Adding '5' gives [tex]\(247 + 5 = 252\)[/tex]

Therefore, the position of the first term where the sum of differences exceeds 250 is at term number [tex]\(27 \times 3 + 3 = 81\)[/tex].

So, the position [tex]\( n \)[/tex] of the first term for which the cumulative sum of the first differences is greater than 250 is [tex]\( 81 \)[/tex].

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