1.1 Consider the following quadratic sequence: …
1.1.1 The first differences also form a sequence. Write down the sequence.
1.1.2 Determine the general term of the first differences.
1.1 .3 Calculate the difference between the and the term
of the quadratic sequence.
1.1.4 If the sum of the first differences is given by .
Calculate the fiftieth of the quadratic sequence.
1.2 (2)
Determine the value of .

Ask by Chang Romero. in South Africa

Mar 23,2025

Question

1.1 Consider the following quadratic sequence: …
1.1.1 The first differences also form a sequence. Write down the sequence.
1.1.2 Determine the general term of the first differences.
1.1 .3 Calculate the difference between the and the term
of the quadratic sequence.
1.1.4 If the sum of the first differences is given by .
Calculate the fiftieth of the quadratic sequence.
1.2 (2)
Determine the value of .

Ask by Chang Romero. in South Africa

Mar 23,2025

UpStudy AI · Accepted Answer

**1.1.1 First differences**

The given sequence is  
$$
-2,\ -3,\ -6,\ -11,\ \dots
$$
We compute the first differences by subtracting consecutive terms:  
- The first difference:  
  $$
  -3 - (-2) = -1
  $$
- The second difference:  
  $$
  -6 - (-3) = -3
  $$
- The third difference:  
  $$
  -11 - (-6) = -5
  $$

Thus, the sequence of first differences is  
$$
-1,\ -3,\ -5,\ \dots
$$

---

**1.1.2 General term of the first differences**

The first differences form an arithmetic sequence. The first term is  
$$
d_1 = -1
$$
and the common difference is  
$$
-3 - (-1) = -2.
$$
Therefore, the general term of an arithmetic sequence is given by  
$$
d_n = d_1 + (n-1)(\text{common difference}).
$$
Substituting the values we get:  
$$
d_n = -1 + (n-1)(-2) = -1 - 2(n-1).
$$
Simplify this expression:  
$$
d_n = -1 - 2n + 2 = -2n + 1.
$$

---

**1.1.3 Difference between the 99th and the 100th term of the quadratic sequence**

For any sequence, the difference between the $$100^\text{th}$$ and the $$99^\text{th}$$ term is the $$99^\text{th}$$ first difference, i.e.,  
$$
T_{100} - T_{99} = d_{99}.
$$
Using the general term for the first differences, we substitute $$n=99$$:  
$$
d_{99} = -2(99) + 1 = -198 + 1 = -197.
$$

---

**1.1.4 Calculate the fiftieth term $$ T_{50} $$ of the quadratic sequence**

We are given that the sum of the first differences is  
$$
S_k=-k^2,
$$
where  
$$
S_k=\sum_{n=1}^{k} d_n.
$$
Notice that, starting from the first term $$ T_1 $$, the $$ (k+1)^\text{th} $$ term is given by  
$$
T_{k+1} = T_1 + S_k.
$$
Since $$ T_1 = -2 $$, the fiftieth term, $$ T_{50} $$, is obtained by taking $$ k = 49 $$:
$$
T_{50} = T_1 + S_{49} = -2 + \big(-49^2\big) = -2 - 2401 = -2403.
$$

---

**1.2 Determine the value of $$ x $$**

There is no additional context or relationship provided in the question that involves $$ x $$. Therefore, there is insufficient information to determine the value of $$ x $$.
**1.1.1 First Differences** The sequence of first differences is $$ -1,\ -3,\ -5,\ \dots $$ --- **1.1.2 General Term of the First Differences** The general term is $$ d_n = -2n + 1 $$ --- **1.1.3 Difference Between the 99th and 100th Terms** The difference is $$ -197 $$ --- **1.1.4 Fiftieth Term $$ T_{50} $$ of the Quadratic Sequence** $$ T_{50} = -2403 $$ --- **1.2 Determine the Value of $$ x $$** Cannot determine the value of $$ x $$ with the given information.

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