[tex]a_1=4[/tex]
[tex]a_2=ra_1=4r[/tex]
[tex]a_3=ra_2=r^2a_1=4r^2[/tex]
...
[tex]a_9=ra_8=\cdots=r^8a_1=4r^8[/tex]
So you have
[tex]1024=4r^8\implies 256=r^8\implies r=2[/tex]
So the sum is
[tex]4+8+16+\cdots+512+1024[/tex]
[tex]4(1+2+4+\cdots+128+256)[/tex]
[tex]4(2^0+2^1+2^2+\cdots+2^7+2^8)[/tex]
[tex]4\displaystyle\sum_{i=1}^92^{i-1}[/tex]
[tex]4\times\dfrac{1-2^9}{1-2}[/tex]
[tex]2044[/tex]
