Determine $$ T_{10} $$ of the following arithmetic sequences: $$ \begin{array}{ll} ext { a) } a=4 ext { and } d=\frac{1}{2} & ext { b) } T_{2}=-10 ext { and } T_{6}=-160 \ ext { d) } T_{5}=8 ext { and } d=3 & ext { d) } T_{2}=4 x ext { and } T_{5}=10 x\end{array} $$

Question

Determine $$ T_{10} $$ of the following arithmetic sequences: $$ \begin{array}{ll}	ext { a) } a=4 	ext { and } d=\frac{1}{2} & 	ext { b) } T_{2}=-10 	ext { and } T_{6}=-160 \ 	ext { d) } T_{5}=8 	ext { and } d=3 & 	ext { d) } T_{2}=4 x 	ext { and } T_{5}=10 x\end{array} $$

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Let's determine the 10th term $$ T_{10} $$ for each of the given arithmetic sequences.

### a) $$ a = 4 $$ and $$ d = \frac{1}{2} $$

**Formula for the $$ n $$-th term:**
$$ T_n = a + (n-1)d $$

**Calculation:**
$$ T_{10} = 4 + (10-1) \times \frac{1}{2} = 4 + 9 \times \frac{1}{2} = 4 + 4.5 = 8.5 $$
$$ T_{10} = \frac{17}{2} $$

---

### b) $$ T_2 = -10 $$ and $$ T_6 = -160 $$

**Set up the equations using the $$ n $$-th term formula:**
$$
\begin{cases}
T_2 = a + d = -10 \
T_6 = a + 5d = -160
\end{cases}
$$

**Subtract the first equation from the second to find $$ d $$:**
$$
(a + 5d) - (a + d) = -160 - (-10) \
4d = -150 \
d = -37.5
$$

**Find $$ a $$ using $$ T_2 = a + d = -10 $$:**
$$
a = -10 - d = -10 - (-37.5) = 27.5
$$

**Now, find $$ T_{10} $$:**
$$
T_{10} = a + 9d = 27.5 + 9 \times (-37.5) = 27.5 - 337.5 = -310
$$

---

### c) $$ T_5 = 8 $$ and $$ d = 3 $$

**Set up the equation using the $$ n $$-th term formula:**
$$
T_5 = a + 4d = 8
$$

**Substitute $$ d = 3 $$:**
$$
a + 12 = 8 \
a = -4
$$

**Now, find $$ T_{10} $$:**
$$
T_{10} = a + 9d = -4 + 9 \times 3 = -4 + 27 = 23
$$

---

### d) $$ T_2 = 4x $$ and $$ T_5 = 10x $$

**Set up the equations using the $$ n $$-th term formula:**
$$
\begin{cases}
T_2 = a + d = 4x \
T_5 = a + 4d = 10x
\end{cases}
$$

**Subtract the first equation from the second to find $$ d $$:**
$$
(a + 4d) - (a + d) = 10x - 4x \
3d = 6x \
d = 2x
$$

**Find $$ a $$ using $$ T_2 = a + d = 4x $$:**
$$
a = 4x - d = 4x - 2x = 2x
$$

**Now, find $$ T_{10} $$:**
$$
T_{10} = a + 9d = 2x + 9 \times 2x = 2x + 18x = 20x
$$

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**Summary of $$ T_{10} $$ for each part:**
- **a)** $$ \frac{17}{2} $$ or $$ 8.5 $$
- **b)** $$ -310 $$
- **c)** $$ 23 $$
- **d)** $$ 20x $$
- **a)** $$ T_{10} = \frac{17}{2} $$ or $$ 8.5 $$ - **b)** $$ T_{10} = -310 $$ - **c)** $$ T_{10} = 23 $$ - **d)** $$ T_{10} = 20x $$

Determine of the following arithmetic sequences:

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