For the sequence, $$ -2,2,6,10, \ldots $$, calculate $$ n $$ if $$ t_{n}=54 $$.

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The given sequence is:
$$
-2, \, 2, \, 6, \, 10, \, \ldots
$$

Let's analyze the pattern:

1. **Identify the type of sequence:**
   - Compute the differences between consecutive terms:
     $$
     2 - (-2) = 4, \quad 6 - 2 = 4, \quad 10 - 6 = 4, \quad \ldots
     $$
   - Since the difference between consecutive terms is constant (4), this is an **arithmetic sequence**.

2. **Determine the general term ($$ t_n $$):**
   - The formula for the $$ n $$-th term of an arithmetic sequence is:
     $$
     t_n = a + (n - 1)d
     $$
     where:
     - $$ a $$ is the first term,
     - $$ d $$ is the common difference.

- Here, $$ a = -2 $$ and $$ d = 4 $$, so:
     $$
     t_n = -2 + (n - 1) \times 4
     $$
     $$
     t_n = -2 + 4n - 4
     $$
     $$
     t_n = 4n - 6
     $$

3. **Find $$ n $$ when $$ t_n = 54 $$:**
   $$
   4n - 6 = 54
   $$
   $$
   4n = 54 + 6
   $$
   $$
   4n = 60
   $$
   $$
   n = \frac{60}{4}
   $$
   $$
   n = 15
   $$

**Answer:** $$ n = 15 $$
$$ n = 15 $$

For the sequence, \( -2,2,6,10, \ldots \), calculate \( n \) if \( t_{n}=54 \).

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