Sure, let's find the first term of the arithmetic sequence given that the common difference [tex]\(d = -3\)[/tex], the 8th term [tex]\(a_8 = 30\)[/tex].
To find the first term [tex]\(a_1\)[/tex], we use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Given:
[tex]\[ d = -3 \][/tex]
[tex]\[ n = 8 \][/tex]
[tex]\[ a_n = 30 \][/tex]
Substituting these values into the formula:
[tex]\[ 30 = a_1 + (8 - 1) \cdot (-3) \][/tex]
[tex]\[ 30 = a_1 + 7 \cdot (-3) \][/tex]
[tex]\[ 30 = a_1 - 21 \][/tex]
To solve for [tex]\(a_1\)[/tex], we add 21 to both sides of the equation:
[tex]\[ a_1 = 30 + 21 \][/tex]
[tex]\[ a_1 = 51 \][/tex]
So, the first term [tex]\(a_1\)[/tex] of the arithmetic sequence is [tex]\(51\)[/tex].
