In Exercises 12–15, write the first six terms of each arithmetic sequence. a1 = 3/2, d = -1/2

Verified step by step guidance

1. Step 1: Recall the formula for the nth term of an arithmetic sequence: $a\_n\; =\; a\_1\; +\; (n\; -\; 1)\; imes\; d$, where $a\_1$ is the first term, $d$ is the common difference, and $n$ is the term number.
2. Step 2: Identify the given values: $a\_1\; =\; \backslash (\backslash frac\{3\}\{2\}\backslash )$ (the first term) and $d\; =\; -\backslash (\backslash frac\{1\}\{2\}\backslash )$ (the common difference).
3. Step 3: Calculate the second term ($a\_2$) using the formula: $a\_2\; =\; a\_1\; +\; d\; =\; \backslash (\backslash frac\{3\}\{2\}\backslash )\; +\; (-\backslash (\backslash frac\{1\}\{2\}\backslash ))$. Simplify the expression to find $a\_2$.
4. Step 4: Calculate the third term ($a\_3$) using the formula: $a\_3\; =\; a\_1\; +\; 2d\; =\; \backslash (\backslash frac\{3\}\{2\}\backslash )\; +\; 2(-\backslash (\backslash frac\{1\}\{2\}\backslash ))$. Simplify the expression to find $a\_3$. Repeat this process for $a\_4$, $a\_5$, and $a\_6$.
5. Step 5: Write the first six terms of the sequence in order: $a\_1,\; a\_2,\; a\_3,\; a\_4,\; a\_5,\; a\_6$, based on the calculations from the previous steps.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference (d). In this case, the first term (a1) is given, and the common difference allows us to generate subsequent terms by adding d to the previous term.

First Term and Common Difference

The first term of an arithmetic sequence is the initial value from which the sequence starts, denoted as a1. The common difference (d) is the fixed amount that is added to each term to obtain the next term. For the given sequence, a1 = 3/2 and d = -1/2, indicating that each term will decrease by 1/2 from the previous term.

Generating Terms of the Sequence

To find the terms of an arithmetic sequence, start with the first term and repeatedly add the common difference. For example, to find the first six terms, calculate each term by applying the formula: a_n = a1 + (n-1)d, where n is the term number. This process allows for systematic generation of the sequence's terms.