#S_n=688color(white)(aaa)a_n=16color(white)(aaa)r=-1/2#
Find #a_1#
First use the formula for a geometric sequence.
#a_n=a_1(r^(n-1))#
#16=a_1(-1/2)^(n-1)#
#a_1=color(red)(frac{16}{(-1/2)^(n-1)})color(white)(aaa)#Equation 1
Next use the formula for the sum of a geometric series.
#S_n=a_1frac{1-r^n}{1-r}#
#688=a_1frac{1- (-1/2)^n}{1- -1/2}#
#688*3/2=a_1(1-(-1/2)^n)#
#688*3/2=color(red)(frac{16}{(-1/2)^(n-1)})*(1-(-1/2)^n)color(white)(aa)#Substitute equation #color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa)# 1 for #a_1#
#688*3/2=frac{16}{(-1/2)^(n-1)}-frac{16*(-1/2)^n}{(-1/2)^(n-1)}#
#688*3/2=frac{16}{(-1/2)^(n-1)}-16(-1/2)^(n-(n-1))#
#688*3/2=frac{16}{(-1/2)^(n-1)}-16(-1/2)#
#1032=frac[16}((-1/2)^(n-10)}+8#
#1024=frac{16}{(-1/2)^(n-1)]#
#(-1/2)^(n-1)=16/1024#
#(-1/2)^(n-1)=1/64#
#(-1/2)^6=1/64#
#n-1=6#
#n=7#
Using #a_n=a_1(r^(n-1))#
#16=a_1(-1/2)^(7-1)#
#16=a_1(1/64)#
#a_1=16*64=1024#
