The first term of the arithmetic sequence is given as [tex]\( a_1 = 1 \)[/tex].
So, [tex]\( a_1 = 1 \)[/tex].
Next, let's calculate the first five terms of the arithmetic sequence with the common difference [tex]\( d = 3 \)[/tex].
1. First term: [tex]\( a_1 = 1 \)[/tex]
2. Second term: This term is found using the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence, which is [tex]\( a_n = a_1 + (n-1)d \)[/tex], where [tex]\( n \)[/tex] is the position of the term in the sequence.
- For the second term ([tex]\( n=2 \)[/tex]):
[tex]\[
a_2 = a_1 + (2-1)d = 1 + 1 \cdot 3 = 1 + 3 = 4
\][/tex]
3. Third term:
- For the third term ([tex]\( n=3 \)[/tex]):
[tex]\[
a_3 = a_1 + (3-1)d = 1 + 2 \cdot 3 = 1 + 6 = 7
\][/tex]
4. Fourth term:
- For the fourth term ([tex]\( n=4 \)[/tex]):
[tex]\[
a_4 = a_1 + (4-1)d = 1 + 3 \cdot 3 = 1 + 9 = 10
\][/tex]
5. Fifth term:
- For the fifth term ([tex]\( n=5 \)[/tex]):
[tex]\[
a_5 = a_1 + (5-1)d = 1 + 4 \cdot 3 = 1 + 12 = 13
\][/tex]
Therefore, the first five terms of the arithmetic sequence with the first term [tex]\( a_1 = 1 \)[/tex] and common difference [tex]\( d = 3 \)[/tex] are:
[tex]\[
1, 4, 7, 10, 13
\][/tex]
