$$ 8 ; 13 ; 18 ; 23 ; \ldots \ldots . $$. is a given sequence. (1) Determine the general rule ( $$ n $$th term (3) Which term is equal to 603 ?

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UpStudy AI · Accepted Answer

To solve the problem, we will first identify the pattern in the given sequence $$ 8, 13, 18, 23, \ldots $$.

### Step 1: Determine the general rule (nth term)

1. **Identify the first term and the common difference:**
   - The first term $$ a_1 = 8 $$.
   - The second term $$ a_2 = 13 $$.
   - The third term $$ a_3 = 18 $$.
   - The fourth term $$ a_4 = 23 $$.

2. **Calculate the common difference:**
   - $$ a_2 - a_1 = 13 - 8 = 5 $$
   - $$ a_3 - a_2 = 18 - 13 = 5 $$
   - $$ a_4 - a_3 = 23 - 18 = 5 $$

The common difference $$ d = 5 $$.

3. **Formulate the nth term:**
   The general formula for the nth term of an arithmetic sequence is given by:
   $$
   a_n = a_1 + (n - 1) \cdot d
   $$
   Substituting the known values:
   $$
   a_n = 8 + (n - 1) \cdot 5
   $$
   Simplifying this:
   $$
   a_n = 8 + 5n - 5 = 5n + 3
   $$

### Step 2: Which term is equal to 603?

To find which term is equal to 603, we set the nth term equal to 603:
$$
5n + 3 = 603
$$

Now, we will solve for $$ n $$:
$$
5n = 603 - 3
$$
$$
5n = 600
$$
$$
n = \frac{600}{5}
$$
$$
n = 120
$$

### Summary of Results
1. The general rule (nth term) is $$ a_n = 5n + 3 $$.
2. The term that is equal to 603 is the 120th term.
The nth term of the sequence is $$ a_n = 5n + 3 $$. The term equal to 603 is the 120th term.

\( 8 ; 13 ; 18 ; 23 ; \ldots \ldots . \). is a given sequence. (1) Determine the general rule ( \( n \)th term (3) Which term is equal to 603 ?

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