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

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

### (a) Determining the 10th Term

In an arithmetic sequence, the difference between consecutive terms is constant. Let's find this common difference $$ d $$.

- The first term ($$ a_1 $$) is $$ x + 3 $$.
- The second term ($$ a_2 $$) is $$ 2x + 6 $$.

Calculating the common difference:
$$
d = a_2 - a_1 = (2x + 6) - (x + 3) = x + 3
$$

The $$ n $$th term of an arithmetic sequence is given by:
$$
a_n = a_1 + (n - 1)d
$$

For the 10th term ($$ a_{10} $$):
$$
a_{10} = (x + 3) + (10 - 1)(x + 3) = (x + 3) + 9(x + 3) = 10(x + 3)
$$

So, the 10th term is:
$$
\boxed{10(x + 3)}
$$

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

Using the same formula for the $$ n $$th term:
$$
a_n = a_1 + (n - 1)d = (x + 3) + (n - 1)(x + 3) = n(x + 3)
$$

Thus, the $$ n $$th term is:
$$
\boxed{n(x + 3)}
$$

Answer:
(a) The tenth term is 10 × (x + 3);
(b) The nth term is n times (x plus three), that is n (x + 3)
(a) The 10th term is $$10(x + 3)$$. (b) The $$n$$th term is $$n(x + 3)$$.

are the first three terms of an arithmetic
sequence.
(a) Determine the 10 th term in terms of .
(b) Determine the th term in terms of .

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are the first three terms of an arithmetic sequence. (a) Determine the 10 th term in terms of . (b) Determine the th term in terms of .