QUESTION 3 3.1 Given the linear pattern: $$ 7 ; 2 ;-3 ; \ldots $$ 3.1.1 $$ \quad $$ Determine the general term, $$ T_{n} $$, of the linear pattern. $$ \begin{array}{l} ext { 3.1.2 } \ \begin{array}{l} ext { 3.1.3 } \ \ 3.2\end{array} \quad ext { Which term in the pattern has a value of }-138 ext { ? } \ ext { Calculate the value of } x\end{array} $$

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QUESTION 3 3.1 Given the linear pattern: $$ 7 ; 2 ;-3 ; \ldots $$ 3.1.1 $$ \quad $$ Determine the general term, $$ T_{n} $$, of the linear pattern. $$ \begin{array}{l}	ext { 3.1.2 } \ \begin{array}{l}	ext { 3.1.3 } \ \ 3.2\end{array} \quad 	ext { Which term in the pattern has a value of }-138 	ext { ? } \ 	ext { Calculate the value of } x\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the general term $$ T_n $$ of the linear pattern

The given linear pattern is: $$ 7, 2, -3, \ldots $$

1. **Identify the first term**: 
   - $$ T_1 = 7 $$

2. **Identify the common difference**:
   - The difference between the first term and the second term is $$ 2 - 7 = -5 $$.
   - The difference between the second term and the third term is $$ -3 - 2 = -5 $$.
   - Thus, the common difference $$ d = -5 $$.

3. **Use the formula for the $$ n $$-th term of an arithmetic sequence**:
   $$
   T_n = T_1 + (n - 1) \cdot d
   $$
   Substituting the known values:
   $$
   T_n = 7 + (n - 1)(-5)
   $$
   Simplifying this:
   $$
   T_n = 7 - 5(n - 1)
   $$
   $$
   T_n = 7 - 5n + 5
   $$
   $$
   T_n = 12 - 5n
   $$

### Step 2: Find which term in the pattern has a value of -138

We need to solve for $$ n $$ when $$ T_n = -138 $$:
$$
12 - 5n = -138
$$

Now, we will solve this equation step by step:

1. **Rearranging the equation**:
   $$
   -5n = -138 - 12
   $$
   $$
   -5n = -150
   $$

2. **Dividing both sides by -5**:
   $$
   n = \frac{-150}{-5} = 30
   $$

### Final Answers

1. The general term $$ T_n $$ of the linear pattern is:
   $$
   T_n = 12 - 5n
   $$

2. The term in the pattern that has a value of -138 is the 30th term:
   $$
   n = 30
   $$
The general term $$ T_n $$ is $$ 12 - 5n $$. The term with a value of -138 is the 30th term.

QUESTION 3
3.1 Given the linear pattern:
3.1.1 Determine the general term, , of the linear pattern.

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QUESTION 3 3.1 Given the linear pattern: 3.1.1 Determine the general term, , of the linear pattern.