$$ x+5 ; 2 x+6 ; 3 x+9 ; \ldots $$ are the first tree terms of an arithmatic sequene. a) Determine the $$ 10^{ ext {th }} $$ term in terms of $$ x $$

Question

$$ x+5 ; 2 x+6 ; 3 x+9 ; \ldots $$ are the first tree terms of an arithmatic sequene. a) Determine the $$ 10^{	ext {th }} $$ term in terms of $$ x $$

UpStudy AI · Accepted Answer

Given the sequence $$ x + 5 ;\ 2x + 6 ;\ 3x + 9 ;\ \ldots $$, we are told that these are the first three terms of an **arithmetic sequence**. An arithmetic sequence has a common difference $$ d $$ between consecutive terms.

Let's denote the first three terms as:
- $$ a_1 = x + 5 $$
- $$ a_2 = 2x + 6 $$
- $$ a_3 = 3x + 9 $$

### Step 1: Determine the Common Difference $$ d $$

For the sequence to be arithmetic, the difference between consecutive terms must be constant. Therefore:

$$
d = a_2 - a_1 = (2x + 6) - (x + 5) = x + 1
$$

$$
d = a_3 - a_2 = (3x + 9) - (2x + 6) = x + 3
$$

Setting these equal for consistency:

$$
x + 1 = x + 3
$$

This simplifies to:

$$
1 = 3
$$

This is a contradiction, indicating that **no such $$ x $$ exists** to make the first three terms form an arithmetic sequence. However, since the problem asks for the 10th term in terms of $$ x $$, we can proceed under the assumption that the sequence is intended to be arithmetic with the common difference $$ d = x + 1 $$.

### Step 2: Find the 10th Term $$ a_{10} $$

The $$ n $$-th term of an arithmetic sequence is given by:

$$
a_n = a_1 + (n - 1)d
$$

Plugging in $$ n = 10 $$, $$ a_1 = x + 5 $$, and $$ d = x + 1 $$:

$$
a_{10} = (x + 5) + (10 - 1)(x + 1) = x + 5 + 9(x + 1) = x + 5 + 9x + 9 = 10x + 14
$$

### Conclusion

Under the assumption that the sequence is arithmetic with a common difference $$ d = x + 1 $$, the 10th term is:

$$
\boxed{10x + 14}
$$
The 10th term is $$ 10x + 14 $$.

are the first
tree terms of an arithmatic sequene.
a) Determine the term in terms of

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