To find the sum of the first 10 terms of the sequence defined by [tex]\( a_n = 4n - 3 \)[/tex], follow these steps:
1. Identify the terms of the sequence: The sequence is given by [tex]\( a_n = 4n - 3 \)[/tex]. We want to find the first 10 terms.
- For [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 4(1) - 3 = 4 - 3 = 1 \)[/tex]
- For [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 4(2) - 3 = 8 - 3 = 5 \)[/tex]
- For [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = 4(3) - 3 = 12 - 3 = 9 \)[/tex]
- For [tex]\( n = 4 \)[/tex]: [tex]\( a_4 = 4(4) - 3 = 16 - 3 = 13 \)[/tex]
- For [tex]\( n = 5 \)[/tex]: [tex]\( a_5 = 4(5) - 3 = 20 - 3 = 17 \)[/tex]
- For [tex]\( n = 6 \)[/tex]: [tex]\( a_6 = 4(6) - 3 = 24 - 3 = 21 \)[/tex]
- For [tex]\( n = 7 \)[/tex]: [tex]\( a_7 = 4(7) - 3 = 28 - 3 = 25 \)[/tex]
- For [tex]\( n = 8 \)[/tex]: [tex]\( a_8 = 4(8) - 3 = 32 - 3 = 29 \)[/tex]
- For [tex]\( n = 9 \)[/tex]: [tex]\( a_9 = 4(9) - 3 = 36 - 3 = 33 \)[/tex]
- For [tex]\( n = 10 \)[/tex]: [tex]\( a_{10} = 4(10) - 3 = 40 - 3 = 37 \)[/tex]
2. List out the first 10 terms:
The first 10 terms are: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37.
3. Calculate the sum of the first 10 terms:
[tex]\[
S_{10} = a_1 + a_2 + a_3 + \cdots + a_{10}
\][/tex]
[tex]\[
S_{10} = 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37
\][/tex]
4. Compute the sum:
[tex]\[
1 + 5 = 6
\][/tex]
[tex]\[
6 + 9 = 15
\][/tex]
[tex]\[
15 + 13 = 28
\][/tex]
[tex]\[
28 + 17 = 45
\][/tex]
[tex]\[
45 + 21 = 66
\][/tex]
[tex]\[
66 + 25 = 91
\][/tex]
[tex]\[
91 + 29 = 120
\][/tex]
[tex]\[
120 + 33 = 153
\][/tex]
[tex]\[
153 + 37 = 190
\][/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].
