What is the explicit formula for the arithmetic sequence –7.5, –9, –10.5, –12, ....? Explain step by step in detail.

To find the explicit formula for an arithmetic sequence, you can use the formula:

${�}_{�}={�}_1+(�-1)�$

Where:

• ${�}_{�}$ is the $�$-th term of the sequence.
• ${�}_1$ is the first term in the sequence.
• $�$ is the position of the term you want to find.
• $�$ is the common difference between terms.

In your case, you have an arithmetic sequence: -7.5, -9, -10.5, -12, ... with a first term (${�}_1$) of -7.5 and a common difference ($�$) of -1.5 (the difference between consecutive terms).

Let's find the explicit formula step by step:

Plug in the known values into the formula:

${�}_{�}=-7.5+(�-1)(-1.5)$

Simplify the formula:

${�}_{�}=-7.5-1.5(�-1)$

Expand the expression:

${�}_{�}=-7.5-1.5�+1.5$

Combine like terms:

${�}_{�}=-6-1.5�$

So, the explicit formula for the arithmetic sequence -7.5, -9, -10.5, -12, ... is: ${�}_{�}=-6-1.5�$

You can use this formula to find any term in the sequence by plugging in the value of $�$. For example, to find the 5th term, substitute $�=5$ into the formula: ${�}_5=-6-1.5\cdot 5=-6-7.5=-13.5$

So, the 5th term is -13.5.