To find the first term [tex]\(a\)[/tex] of the geometric sequence, we will use the information provided and the properties of geometric sequences.
A geometric sequence has the form:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
We are given:
[tex]\[ a_7 = 64 \][/tex]
[tex]\[ r = -2 \][/tex]
We need to find the first term [tex]\(a\)[/tex].
Using the formula for the [tex]\(n\)[/tex]th term:
[tex]\[ a_7 = a \cdot r^{7-1} \][/tex]
[tex]\[ a_7 = a \cdot r^6 \][/tex]
Substituting the given values into the formula:
[tex]\[ 64 = a \cdot (-2)^6 \][/tex]
We know that:
[tex]\[ (-2)^6 = 64 \][/tex]
So the equation becomes:
[tex]\[ 64 = a \cdot 64 \][/tex]
To isolate [tex]\(a\)[/tex], we divide both sides of the equation by 64:
[tex]\[ a = \frac{64}{64} \][/tex]
[tex]\[ a = 1.0 \][/tex]
Thus, the first term of the geometric sequence is:
[tex]\[ a = 1.0 \][/tex]
