Answer:
The sum of the first 9 terms is [tex]S_9=\frac{511}{4}[/tex].
Step-by-step explanation:
To find the sum of the first [tex]S_n[/tex] terms of a geometric sequence use the formula
[tex]S_n=\frac{a_1(1-r^n)}{1-r}[/tex]
where, [tex]n[/tex] is the number of terms, [tex]a_1[/tex] is the first term and [tex]r[/tex] is the common ratio.
To find the common ratio, find the ratio between a term and the term preceding it.
Given the geometric sequence [tex]64+32+16+...[/tex], the common ratio is
[tex]r=\frac{32}{64} =\frac{1}{2}[/tex]
and the sum of the first 9 terms is
[tex]S_9=\frac{64(1-(\frac{1}{2})^9)}{1-\frac{1}{2}}\\\\S_9=\frac{64\left(-\left(\frac{1}{2}\right)^9+1\right)}{\frac{1}{2}}\\\\S_9=\frac{64\left(-\frac{1}{2^9}+1\right)}{\frac{1}{2}}\\\\S_9=\frac{64\left(-\frac{1}{512}+1\right)}{\frac{1}{2}}\\\\S_9=\frac{64\left(1-\frac{1}{512}\right)\cdot \:2}{1}=\frac{128\left(-\frac{1}{512}+1\right)}{1}\\\\S_9=\frac{128\cdot \frac{511}{512}}{1}=\frac{511\cdot \:128}{512}=\frac{65408}{512}=\frac{511}{4}[/tex]
Answer: 127.75.
(Got it wrong, clicked on help, and it showed me the answer. Done.)
