3 Mathematics/control test 1 SBA task 2 Ehlanzeni QUESTION 1 1.1 Consider the following quadratic sequence: $$ -2 ;-3 ;-6 ;-11 ; \ldots $$ 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 $$ 99^{\mathrm{th}} $$ and the $$ 100^{\mathrm{dh}} $$ term of the quadratic sequence. 1.1.4 If the sum of the first differences is given by $$ S_{k}=-k^{2} $$. Calculate the fiftieth ( $$ T_{50} $$ ) of the quadratic sequence. $$ 1.2 \quad 4-x ; 3 $$ and $$ \sqrt{2 x+3} $$ are the first three terms of an arithmetic sequence. Determine the value of $$ x $$.

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**1.1 Given the quadratic sequence:**  
$$
-2,\; -3,\; -6,\; -11,\; \ldots
$$
with
$$
T_{1}=-2,\quad T_{2}=-3,\quad T_{3}=-6,\quad T_{4}=-11,\quad \ldots
$$

**1.1.1 First differences**  
The differences between consecutive terms are calculated as follows:
$$
T_{2}-T_{1}=-3-(-2)=-1,
$$
$$
T_{3}-T_{2}=-6-(-3)=-3,
$$
$$
T_{4}-T_{3}=-11-(-6)=-5.
$$
Thus, the sequence of first differences is:
$$
-1,\; -3,\; -5,\; \ldots
$$

---

**1.1.2 General term of the first differences**  
The sequence of first differences is an arithmetic sequence with first term 
$$
d_{1} = -1
$$
and common difference 
$$
\Delta d = -3 - (-1) = -2.
$$
The general term of an arithmetic sequence is given by:
$$
d_{n} = d_{1} + (n-1)(\Delta d).
$$
Substitute the given values:
$$
d_{n}=-1+(n-1)(-2)=-1-2(n-1)=-2n+1.
$$
Thus, the general term of the first differences is:
$$
d_{n}=-2n+1.
$$

---

**1.1.3 Difference between the $$ 99^{\mathrm{th}} $$ and $$ 100^{\mathrm{th}} $$ term of the quadratic sequence**  
The difference $$ T_{100}-T_{99} $$ is the $$ 99^{\mathrm{th}} $$ term in the first differences sequence (since the difference between $$ T_{n+1} $$ and $$ T_{n} $$ corresponds to $$ d_{n} $$). Therefore:
$$
T_{100}-T_{99}=d_{99}=-2(99)+1=-198+1=-197.
$$

---

**1.1.4 Calculate the $$ 50^{\mathrm{th}} $$ term of the quadratic sequence**  
We are given that the sum of the first differences is:
$$
S_{k}=-k^{2}.
$$
Recall that the $$ n^{\mathrm{th}} $$ term of the original sequence can be found by:
$$
T_{n} = T_{1} + S_{n-1}.
$$
For $$ T_{50} $$, we have:
$$
T_{50} = T_{1} + S_{49}.
$$
Given $$ T_{1}=-2 $$ and $$ S_{49} = -(49)^2 = -2401 $$, we obtain:
$$
T_{50} = -2 - 2401 = -2403.
$$

---

**1.2 Determine the value of $$ x $$ in the arithmetic sequence**

The first three terms are:
$$
4-x,\quad 3,\quad \sqrt{2x+3}.
$$
For these to form an arithmetic sequence, the common difference must be the same. Therefore:
$$
3 - (4-x) = \sqrt{2x+3} - 3.
$$
Calculate the left-hand side:
$$
3 - (4-x) = 3-4+x = x-1.
$$
Thus, the equation becomes:
$$
x-1=\sqrt{2x+3}-3.
$$
Isolate the square root:
$$
\sqrt{2x+3}= x-1+3 = x+2.
$$
Since $$\sqrt{2x+3} \ge 0$$, we require:
$$
x+2 \ge 0 \quad \Longrightarrow \quad x\ge -2.
$$
Now, square both sides:
$$
2x+3=(x+2)^2=x^2+4x+4.
$$
Bring all terms to one side:
$$
0=x^2+4x+4-2x-3=x^2+2x+1.
$$
Notice that:
$$
x^2+2x+1=(x+1)^2.
$$
Thus,
$$
(x+1)^2=0 \quad \Longrightarrow \quad x=-1.
$$
Verification:  
For $$ x=-1 $$,
$$
4-x=4-(-1)=5,\quad \text{and} \quad \sqrt{2(-1)+3}=\sqrt{-2+3}=\sqrt{1}=1.
$$
The three terms are:
$$
5,\; 3,\; 1,
$$
which form an arithmetic sequence with common difference:
$$
3-5=-2,\quad 1-3=-2.
$$
Thus, the value of $$ x $$ is:
$$
x=-1.
$$
**1.1 Quadratic Sequence Analysis** 1. **First Differences:** - Sequence: $$-1, -3, -5, \ldots$$ 2. **General Term of First Differences:** - $$ d_n = -2n + 1 $$ 3. **Difference Between 99th and 100th Terms:** - $$ T_{100} - T_{99} = -197 $$ 4. **50th Term of Quadratic Sequence:** - $$ T_{50} = -2403 $$ **1.2 Arithmetic Sequence Determination** - **Value of $$ x $$:** - $$ x = -1 $$

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