To find the sum of the first 10 terms of the sequence defined by the formula [tex]\(a_n = 4n - 3\)[/tex], we can follow these steps:
1. Identify the formula for the [tex]\(n\)[/tex]-th term of the sequence: [tex]\(a_n = 4n - 3\)[/tex].
2. Calculate the first 10 terms of the sequence:
- For [tex]\(n=1\)[/tex], [tex]\(a_1 = 4 \cdot 1 - 3 = 1\)[/tex]
- For [tex]\(n=2\)[/tex], [tex]\(a_2 = 4 \cdot 2 - 3 = 5\)[/tex]
- For [tex]\(n=3\)[/tex], [tex]\(a_3 = 4 \cdot 3 - 3 = 9\)[/tex]
- For [tex]\(n=4\)[/tex], [tex]\(a_4 = 4 \cdot 4 - 3 = 13\)[/tex]
- For [tex]\(n=5\)[/tex], [tex]\(a_5 = 4 \cdot 5 - 3 = 17\)[/tex]
- For [tex]\(n=6\)[/tex], [tex]\(a_6 = 4 \cdot 6 - 3 = 21\)[/tex]
- For [tex]\(n=7\)[/tex], [tex]\(a_7 = 4 \cdot 7 - 3 = 25\)[/tex]
- For [tex]\(n=8\)[/tex], [tex]\(a_8 = 4 \cdot 8 - 3 = 29\)[/tex]
- For [tex]\(n=9\)[/tex], [tex]\(a_9 = 4 \cdot 9 - 3 = 33\)[/tex]
- For [tex]\(n=10\)[/tex], [tex]\(a_{10} = 4 \cdot 10 - 3 = 37\)[/tex]
3. Sum these first 10 terms:
[tex]\[
a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 + a_{10} = 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37
\][/tex]
4. Adding these terms together we get:
[tex]\[
1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37 = 190
\][/tex]
Therefore, the sum of the first 10 terms of the sequence defined by [tex]\(a_n = 4n - 3\)[/tex] is [tex]\( \boxed{190} \)[/tex].
