To find the sum of the first 10 terms of the given arithmetic sequence [tex]\(\{-3, 2, 7, 12, 17, 22, \ldots\}\)[/tex], we can use the formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence:
[tex]\[ S_n = \frac{1}{2} n (2a + (n-1)d) \][/tex]
where:
- [tex]\( S_n \)[/tex] is the sum of the first [tex]\(n\)[/tex] terms.
- [tex]\( n \)[/tex] is the number of terms.
- [tex]\( a \)[/tex] is the first term.
- [tex]\( d \)[/tex] is the common difference between consecutive terms.
Let's identify the values from the given sequence:
- The first term [tex]\(a\)[/tex] is [tex]\(-3\)[/tex].
- The common difference [tex]\(d\)[/tex] is calculated by subtracting the first term from the second term: [tex]\(d = 2 - (-3) = 5\)[/tex].
- The number of terms [tex]\(n\)[/tex] is 10.
Now, plug these values into the formula:
[tex]\[ S_{10} = \frac{1}{2} \times 10 \times (2 \times (-3) + (10 - 1) \times 5) \][/tex]
Next, calculate the expression inside the parentheses:
[tex]\[ 2 \times (-3) + (10 - 1) \times 5 = -6 + 9 \times 5 \][/tex]
[tex]\[ -6 + 45 = 39 \][/tex]
Now substitute back into the formula:
[tex]\[ S_{10} = \frac{1}{2} \times 10 \times 39 \][/tex]
[tex]\[ S_{10} = 5 \times 39 \][/tex]
[tex]\[ S_{10} = 195 \][/tex]
Therefore, the sum of the first 10 terms of the sequence is [tex]\(195\)[/tex].
Answer: [tex]\( 195 \)[/tex]
