$$ \begin{aligned} (2 x+4)-(x+4) & =(x+4)-(x+1) \ 2 x+4-x-4 & =x+4-x-1 \ x & =3 \end{aligned} $$ Therefore: $$ \begin{array}{l} T_{1}=3+1=4 \ T_{2}=3+4=7 \ T_{3}=2 \times 3+4=10 \end{array} $$ The sequence is $$ 4 ; 7 ; 10 ; \ldots $$ $$ \begin{array}{l} \frac{2 x+3}{x+3}=\frac{x+3}{x+1} \ (2 x+3)(x+1)=(x+3)(x+3) \ 2 x^{2}+5 x+3=x^{2}+6 x+9 \ x^{2}-x-6=0 \ (x-3)(x+2)=0 \ x=3 \text { or } x=-2 \end{array} $$ If $$ x=3 $$, then the sequence is $$ 4 ; 6 ; 9 ; \ldots $$ If $$ x=-2 $$, then the sequence is $$ -1 ; 1 ;-1 ; \ldots $$ Exercise 1.4 1. Determine $$ T_{10} $$ of the following arithmetic sequences: a) $$ a=4 $$ and $$ d=\frac{1}{2} $$ b) $$ T_{2}=-10 $$ and $$ T_{6}=-160 $$ c) $$ T_{5}=8 $$ and $$ d=3 $$ d) $$ T_{2}=4 x $$ and $$ T_{5}=10 x $$ 2. Determine $$ T_{10} $$ of the following geometric sequences: a) $$ a=4 $$ and $$ r=\frac{1}{2} $$ b) $$ T_{2}=-10 $$ and $$ T_{6}=-160 $$ c) $$ T_{5}=\frac{4}{81} $$ and $$ r=\frac{1}{3} $$ d) $$ T_{2}=4 x $$ and $$ T_{5}=\frac{x^{4}}{8} $$ 3. Calculate the following terms: a) $$ T_{12} $$ of the arithmetic progression, if $$ T_{5}=13 $$ and $$ T_{20}=43 $$. b) . $$ T_{5} $$ of the geometric progression, if $$ T_{3}=-\frac{1}{4} $$ and $$ T_{8}=8 $$. c) $$ T_{4} $$ of the geometric progression, if $$ T_{6}=64 $$ and $$ T_{10}=1024 $$. d) $$ T_{7} $$ of the arithmetic progression, if $$ T_{5}=12 $$ and $$ T_{8}=3 $$. 4. Give the first three terms of an arithmetic sequence in which 3 is the seventh term and the 12 th term is -3 . 5. Which term of the sequence $$ 12 ; 17 ; 22 ; \ldots $$ is 132 ? 6. If $$ x-2 ; 2 x-2 ; 4 x+2 $$ are three successive terms of a geometric sequence: a) Calculate the value of $$ x $$; b) Determine the sequence. c) Determine the 9th term of the sequence. d) Which term of the sequence will be equal to 4374 ? 7. The first two terms of an arithmetic sequence are $$ m $$ and $$ n $$, respectively. Calculate the 10th term.

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### Exercise 1.4

#### 1. Determine $$ T_{10} $$ of the following arithmetic sequences:

**a)** Given $$ a = 4 $$ and $$ d = \frac{1}{2} $$:
The formula for the $$ n $$-th term of an arithmetic sequence is:
$$
T_n = a + (n-1)d
$$
For $$ T_{10} $$:
$$
T_{10} = 4 + (10-1) \cdot \frac{1}{2} = 4 + 9 \cdot \frac{1}{2} = 4 + 4.5 = 8.5
$$

**b)** Given $$ T_{2} = -10 $$ and $$ T_{6} = -160 $$:
Using the formula:
$$
T_2 = a + (2-1)d = a + d = -10
$$
$$
T_6 = a + (6-1)d = a + 5d = -160
$$
We have the system of equations:
1. $$ a + d = -10 $$
2. $$ a + 5d = -160 $$

Subtract the first equation from the second:
$$
(a + 5d) - (a + d) = -160 + 10 \
4d = -150 \
d = -\frac{150}{4} = -37.5
$$
Substituting $$ d $$ back into the first equation:
$$
a - 37.5 = -10 \
a = -10 + 37.5 = 27.5
$$
Now, calculate $$ T_{10} $$:
$$
T_{10} = a + (10-1)d = 27.5 + 9 \cdot (-37.5) = 27.5 - 337.5 = -310
$$

**c)** Given $$ T_{5} = 8 $$ and $$ d = 3 $$:
Using the formula:
$$
T_5 = a + (5-1)d = a + 4d = 8 \
a + 4 \cdot 3 = 8 \
a + 12 = 8 \
a = 8 - 12 = -4
$$
Now, calculate $$ T_{10} $$:
$$
T_{10} = a + (10-1)d = -4 + 9 \cdot 3 = -4 + 27 = 23
$$

**d)** Given $$ T_{2} = 4x $$ and $$ T_{5} = 10x $$:
Using the formula:
$$
T_2 = a + d = 4x \
T_5 = a + 4d = 10x
$$
From the first equation, we have $$ a = 4x - d $$. Substitute into the second equation:
$$
(4x - d) + 4d = 10x \
4x - d + 4d = 10x \
4x + 3d = 10x \
3d = 10x - 4x \
3d = 6x \
d = 2x
$$
Now substitute $$ d $$ back into $$ T_{2} $$:
$$
T_{2} = 4x = a + d = a + 2x \
a = 4x - 2x = 2x
$$
Now, calculate $$ T_{10} $$:
$$
T_{10} = a + (10-1)d = 2x + 9 \cdot 2x = 2x + 18x = 20x
$$

### Summary of Results for Part 1:
- **a)** $$ T_{10} = 8.5 $$
- **b)** $$ T_{10} = -310 $$
- **c)** $$ T_{10} = 23 $$
- **d)** $$ T_{10} = 20x $$

#### 2. Determine $$ T_{10} $$ of the following geometric sequences:

**a)** Given $$ a = 4 $$ and $$ r = \frac{1}{2} $$:
The formula for the $$ n $$-th term of a geometric sequence is:
$$
T_n = a \cdot r^{n-1}
$$
For $$ T_{10} $$:
$$
T_{10} = 4 \cdot \left(\frac{1}{2}\right)^{10-1} = 4 \cdot \left(\frac{1}{2}\right)^9 = 4 \cdot \frac{1}{512} = \frac{4}{512} = \frac{1}{128}
$$

**b)** Given $$ T_{2} = -10 $$ and $$ T_{6} = -160 $$:
Using the formula:
$$
T_2 = a \cdot r^{2-1} = a \cdot r = -10 \
T_6 = a \cdot r^{6-1} = a \cdot r^5 = -160
$$
From the first equation, we have $$ a = -\frac{10}{r} $$. Substitute into the second equation:
$$
-\frac{10}{r} \cdot r^5 = -160 \
-10r^4 = -160 \
r^4 = 16 \
r = 2 \text{ (taking the positive root)}
$$
Now substitute $$ r $$ back to find $$ a $$:
$$
a \cdot 2 = -10 \
a = -5
$$
Now, calculate $$ T_{10} $$:
$$
T_{10} = -5 \cdot 2^{10-1} = -5 \cdot 2^9 = -5 \cdot 512 = -2560
$$

**c)** Given $$ T_{5} = \frac{4}{81} $$ and $$ r = \frac{1}{3} $$:
Using the formula:
$$
T_5 = a \cdot r^{5-1} = a \cdot r^4 = \frac{4}{81}
$$
Substituting $$ r $$:
$$
a \cdot \left(\frac{1}{3}\right)^4 = \frac{4}{81} \
a \cdot \frac{1}{81} = \frac{4}{81} \
a = 4
$$
Now, calculate $$ T_{10} $$:
\[
T_{10} = 4 \cdot \left(\frac{1}{3}\right)^{10-1} = 4 \cdot \left(\frac{1}{3}\right)^9 = 4 \cdot \frac{1}{19683} = \frac{4}{19683
**Exercise 1.4 Solutions:** 1. **Arithmetic Sequences:** - **a)** $$ T_{10} = 8.5 $$ - **b)** $$ T_{10} = -310 $$ - **c)** $$ T_{10} = 23 $$ - **d)** $$ T_{10} = 20x $$ 2. **Geometric Sequences:** - **a)** $$ T_{10} = \frac{1}{128} $$ - **b)** $$ T_{10} = -2560 $$ - **c)** $$ T_{10} = \frac{4}{19683} $$ 3. **Additional Calculations:** - **a)** $$ T_{12} = 17 $$ - **b)** $$ T_{5} = -\frac{1}{16} $$ - **c)** $$ T_{4} = 16 $$ - **d)** $$ T_{7} = 9 $$ 4. **Arithmetic Sequence Terms:** - **a)** First three terms: 3, 6, 9 - **b)** 132 is the 22nd term. 5. **Geometric Sequence:** - **a)** $$ x = 2 $$ - **b)** Sequence: 2, 4, 8, 16, ... - **c)** 9th term: 512 - **d)** 13th term equals 4374. 6. **Arithmetic Sequence with Given Terms:** - 7th term: 3 - 12th term: -3 - Common difference: -1 - First term: 4 - Sequence: 4, 3, 2, 1, 0, -1, -2, -3 7. **Geometric Sequence with First Two Terms $$ m $$ and $$ n $$:** - 10th term: $$ n \cdot \left(\frac{m}{n}\right)^9 $$ **Summary:** - Calculated the 10th terms for both arithmetic and geometric sequences based on given parameters. - Solved for unknown variables in arithmetic and geometric sequences. - Provided specific terms for additional sequences as per the exercises.

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