3.1 Given the linear pattern: $$ 7 ; 2 ;-3 ; \ldots $$ 3.1.1 Determine the general term, $$ T_{n} $$, of the linear pattern. 3.1.2 Calculate the value of $$ T_{20} $$. 3.1.3 Which term in the pattern has a value of -138 ?

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**Step 1. Find the general term $$ T_n $$ for the sequence**

The pattern is given by the terms:
$$
7,\quad 2,\quad -3,\quad \ldots
$$
The common difference $$ d $$ is calculated as:
$$
d = 2 - 7 = -5
$$
A linear (arithmetic) sequence has the general term:
$$
T_n = a + (n-1)d
$$
where $$ a $$ is the first term.

Substituting $$ a = 7 $$ and $$ d = -5 $$:
$$
T_n = 7 + (n-1)(-5)
$$
Expanding the expression:
$$
T_n = 7 - 5(n-1)
$$
$$
T_n = 7 - 5n + 5
$$
$$
T_n = 12 - 5n
$$

---

**Step 2. Calculate $$ T_{20} $$**

Using the general formula $$ T_n = 12 - 5n $$, substitute $$ n = 20 $$:
$$
T_{20} = 12 - 5(20)
$$
$$
T_{20} = 12 - 100
$$
$$
T_{20} = -88
$$

---

**Step 3. Determine the term number for which the value is $$-138$$**

We need to find $$ n $$ such that:
$$
T_n = -138
$$
Substitute the formula $$ T_n = 12 - 5n $$:
$$
12 - 5n = -138
$$
Subtract 12 from both sides:
$$
-5n = -138 - 12
$$
$$
-5n = -150
$$
Divide by $$-5$$:
$$
n = \frac{-150}{-5}
$$
$$
n = 30
$$

---

**Final Answers:**

1. The general term is:  
   $$
   T_n = 12 - 5n
   $$

2. The 20th term is:  
   $$
   T_{20} = -88
   $$

3. The term with the value $$-138$$ is the 30th term.
**General Term:** $$ T_n = 12 - 5n $$ **20th Term:** $$ T_{20} = -88 $$ **Term with Value -138:** 30th term

3.1 Given the linear pattern:
3.1.1 Determine the general term, , of the linear pattern.
3.1.2 Calculate the value of .
3.1.3 Which term in the pattern has a value of -138 ?

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3.1 Given the linear pattern: 3.1.1 Determine the general term, , of the linear pattern. 3.1.2 Calculate the value of . 3.1.3 Which term in the pattern has a value of -138 ?