To solve this question, we'll identify the common difference ([tex]\(d\)[/tex]) for the arithmetic sequence and use it to complete the table. Here are the provided terms:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_3 = -2 \)[/tex]
- [tex]\( a_{10} = -23 \)[/tex]
### Step-by-Step Solution:
1. Determine the Common Difference:
- The [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by [tex]\( a_n = a_1 + (n-1)d \)[/tex].
2. Calculate [tex]\(d\)[/tex] using terms 1 and 3:
- For [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = a_1 + 2d \)[/tex]
- Substituting the known values: [tex]\( -2 = 4 + 2d \)[/tex]
- Solving for [tex]\(d\)[/tex]: [tex]\( 2d = -2 - 4 \)[/tex]
- [tex]\( d = -6 / 2 \)[/tex]
- [tex]\( d = -3 \)[/tex]
3. Validate the common difference using term 10:
- For [tex]\( n = 10 \)[/tex]: [tex]\( a_{10} = a_1 + 9d \)[/tex]
- Substituting the known values: [tex]\( -23 = 4 + 9d \)[/tex]
- Solving for [tex]\(d\)[/tex]: [tex]\( 9d = -23 - 4 \)[/tex]
- [tex]\( 9d = -27 \)[/tex]
- [tex]\( d = -3 \)[/tex]
So, the common difference [tex]\( d \)[/tex] is confirmed as [tex]\( -3 \)[/tex].
4. Complete the table:
- For [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 4 \)[/tex]
- For [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = a_1 + d = 4 + (-3) = 1 \)[/tex]
- For [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = -2 \)[/tex]
- For [tex]\( n = 4 \)[/tex]: [tex]\( a_4 = a_3 + d = -2 + (-3) = -5 \)[/tex]
- For [tex]\( n = 5 \)[/tex]: [tex]\( a_5 = a_4 + d = -5 + (-3) = -8 \)[/tex]
- For [tex]\( n = 6 \)[/tex]: [tex]\( a_6 = a_5 + d = -8 + (-3) = -11 \)[/tex]
- For [tex]\( n = 7 \)[/tex]: [tex]\( a_7 = a_6 + d = -11 + (-3) = -14 \)[/tex]
- For [tex]\( n = 8 \)[/tex]: [tex]\( a_8 = a_7 + d = -14 + (-3) = -17 \)[/tex]
- For [tex]\( n = 9 \)[/tex]: [tex]\( a_9 = a_8 + d = -17 + (-3) = -20 \)[/tex]
- For [tex]\( n = 10 \)[/tex]: [tex]\( a_{10} = -23 \)[/tex]
Thus, the complete table is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Term \# }(n) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\text{Term }(a_n) & 4 & 1 & -2 & -5 & -8 & -11 & -14 & -17 & -20 & -23 \\
\hline
\end{array}
\][/tex]
In this arithmetic sequence:
- The common difference [tex]\( d \)[/tex] is [tex]\( -3 \)[/tex].
- The terms sequence: [tex]\( 4, 1, -2, -5, -8, -11, -14, -17, -20, -23 \)[/tex].
