Answer:
6 terms
Step-by-step explanation:
we have
[tex]2+8+32+..+2048[/tex]
we need to identify a pattern so
from 2 to 8 what could have happened?
well you could have added 6 and get 8
so that means that to 8 you have to add 6 and get 32
WRONG
so adding is not an option
what else could have happened?
well you could have multiplied by 4 and get 8
so that means that to 8 you have to multiply by 4 and get 32
which is true
so the geometric series is
[tex]2+4(2)+4(4(2))+4(4(4(2)))+...+[/tex] what?
you can see that in the 2nd term we have 1 four, in the 3rd we have 2 fours
so we can conclude in the nth term we have n-1 fours
so the formula is
[tex]2+8+32+...+4^{n-1} *2[/tex]
so we need to determine when
[tex]4^{n-1}*2=2048[/tex]
[tex]4^{n-1}*2*\frac{1}{2} =2048*\frac{1}{2} \\\\4^{n-1}=1024[/tex]
if we factorize 4 as [tex]2^2[/tex] and 1024 as [tex]2^{10}[/tex]
we have
[tex](2^2)^{n-1}=2^{10}\\\\2^{2(n-1)}=2^{10}\\\\2^{2n-2}=2^{10}[/tex]
so we see that the bases are equal, that must mean the exponents are equal
[tex]2n-2=10\\\\2n=10+2\\\\2n=12\\\\n=6[/tex]
so there are 6 terms
