Let's determine the first six terms of the sequence defined by the formula [tex]\( a_n = 2^n - 1 \)[/tex].
1. First Term ([tex]\(a_1\)[/tex]):
[tex]\[
a_1 = 2^1 - 1 = 2 - 1 = 1
\][/tex]
2. Second Term ([tex]\(a_2\)[/tex]):
[tex]\[
a_2 = 2^2 - 1 = 4 - 1 = 3
\][/tex]
3. Third Term ([tex]\(a_3\)[/tex]):
[tex]\[
a_3 = 2^3 - 1 = 8 - 1 = 7
\][/tex]
4. Fourth Term ([tex]\(a_4\)[/tex]):
[tex]\[
a_4 = 2^4 - 1 = 16 - 1 = 15
\][/tex]
5. Fifth Term ([tex]\(a_5\)[/tex]):
[tex]\[
a_5 = 2^5 - 1 = 32 - 1 = 31
\][/tex]
6. Sixth Term ([tex]\(a_6\)[/tex]):
[tex]\[
a_6 = 2^6 - 1 = 64 - 1 = 63
\][/tex]
Hence, the first six terms of the sequence [tex]\(a_n = 2^n - 1\)[/tex] are:
[tex]\[ 1, 3, 7, 15, 31, 63 \][/tex]
