Answer:
1023
Step-by-step explanation:
[tex]8^{0}[/tex] = 1, [tex]8^{\frac{1}{3} }[/tex] = [tex]\sqrt[3]{8}[/tex] = 2, [tex]8^{\frac{2}{3} }[/tex] = [tex]\sqrt[3]{8^2}[/tex] = 4, [tex]8^{1}[/tex] = 8
The series can be expressed in non exponent terms as
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 ← 10 terms
There is a common ratio r between consecutive terms, that is
r = 2 ÷ 1 = 4 ÷ 2 = 8 ÷ 4 = .... = 2
This indicates the sequence is geometric with sum to n terms
[tex]S_{n}[/tex] = [tex]\frac{a(r^{n}-1) }{r-1}[/tex]
where a is the first term and r the common ratio
Here a = 1 and r = 2, thus
[tex]S_{10}[/tex] = [tex]\frac{1(2^{10}-1) }{2-1}[/tex] = 1024 - 1 = 1023
