For a geometric sequence with common ratio [tex]r[/tex], we have
[tex]a_n=a_{n-1}r[/tex]
that is, the [tex]n[/tex]-th term in the sequence is the product of the previous term and the common ratio [tex]r[/tex]. So
[tex]a_n=a_{n-1}r=a_{n-2}r^2=\cdots=a_1r^{n-1}[/tex]
Then the sum of the first 7 terms is
[tex]S_7=\displaystyle\sum_{n=1}^7a_n=a_1+a_2+\cdots+a_7[/tex]
[tex]\implies S_7=a_1+a_1r+\cdots+a_1r^6[/tex]
Notice that
[tex]S_7r=a_1r+a_1r^2+\cdots+a_1r^7[/tex]
so we can subtract this modified sum from [tex]S_7[/tex] to get
[tex]S_7-S_7r=S_7(1-r)=a_1-a_1r^7\implies S_7=a_1\dfrac{1-r^7}{1-r}[/tex]
We're told that [tex]a_1=-11[/tex] and [tex]r=-4[/tex], so the sum of the first 7 terms is
[tex]S_7=(-11)\dfrac{1-(-4)^7}{1-(-4)}=144,188[/tex]
