$$ x+3 ; 2 x+6 ; 3 x+9 ; \ldots $$ are the first three terms of an arithmetic sequence. (a) Determine the 10 th term in terms of $$ x $$. (b) Determine the $$ n $$th term in terms of $$ x $$.

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Let's analyze the given arithmetic sequence:

$$
x + 3,\; 2x + 6,\; 3x + 9,\; \ldots
$$

### (a) Determining the 10th Term

In an arithmetic sequence, each term increases by a common difference $$ d $$.

1. **First Term ($$ a $$)**:
   $$
   a = x + 3
   $$

2. **Common Difference ($$ d $$)**:
   $$
   d = (2x + 6) - (x + 3) = x + 3
   $$

3. **10th Term ($$ T_{10} $$)**:
   The $$ n $$th term of an arithmetic sequence is given by:
   $$
   T_n = a + (n-1)d
   $$
   Plugging in $$ n = 10 $$:
   $$
   T_{10} = (x + 3) + 9(x + 3) = x + 3 + 9x + 27 = 10x + 30
   $$

### (b) Determining the $$ n $$th Term

Using the general formula for the $$ n $$th term:

$$
T_n = a + (n-1)d
$$

Plugging in $$ a = x + 3 $$ and $$ d = x + 3 $$:

$$
T_n = (x + 3) + (n-1)(x + 3) = (1 + n - 1)x + 3 + 3(n - 1) = nx + 3n
$$

Alternatively, it can be factored as:

$$
T_n = n(x + 3)
$$

### **Final Answers**

**(a)** The 10th term is $$ 10x + 30 $$.

**(b)** The nth term is $$ n\,x + 3\,n $$.
**(a)** The 10th term is $$ 10x + 30 $$. **(b)** The nth term is $$ n(x + 3) $$.

\( x+3 ; 2 x+6 ; 3 x+9 ; \ldots \) are the first three terms of an arithmetic sequence. (a) Determine the 10 th term in terms of \( x \). (b) Determine the \( n \)th term in terms of \( x \).

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