Answer:
The sum of the first 8 terms will be:
Step-by-step explanation:
Given the sequence
[tex]36, 30, 25, ...[/tex]
A geometric sequence has a constant ratio and is defined by
[tex]a_n=a_0\cdot r^{n-1}[/tex]
computing the common ratio
[tex]\frac{30}{36}=\frac{5}{6},\:\quad \frac{25}{30}=\frac{5}{6}[/tex]
[tex]r=\frac{5}{6}[/tex]
As the first element is
[tex]a_1=36[/tex]
so the nth term will be:
[tex]a_n=a_0\cdot r^{n-1}[/tex]
[tex]a_n=36\left(\frac{5}{6}\right)^{n-1}[/tex]
Geometric sequence sum formula
[tex]a_1\frac{1-r^n}{1-r}[/tex]
Plugin the values to determine the sum of the first 8 terms
[tex]n=8,\:\spacea_1=36,\:\spacer=\frac{5}{6}[/tex]
[tex]=36\cdot \frac{1-\left(\frac{5}{6}\right)^8}{1-\frac{5}{6}}[/tex]
[tex]=\frac{\left(1-\left(\frac{5}{6}\right)^8\right)\cdot \:36}{1-\frac{5}{6}}[/tex]
[tex]=\frac{36\left(-\left(\frac{5}{6}\right)^8+1\right)}{\frac{1}{6}}[/tex]
[tex]=\frac{36\left(-\frac{390625}{1679616}+1\right)}{\frac{1}{6}}[/tex]
[tex]=\frac{\left(1-\frac{390625}{1679616}\right)\cdot \:36\cdot \:6}{1}[/tex]
[tex]=\frac{216\left(-\frac{390625}{1679616}+1\right)}{1}[/tex]
[tex]=\frac{216\cdot \frac{1288991}{1679616}}{1}[/tex]
[tex]=216\cdot \frac{1288991}{1679616}[/tex]
[tex]=\frac{1288991\cdot \:216}{1679616}[/tex]
[tex]=\frac{278422056}{1679616}[/tex]
[tex]=\frac{1288991}{7776}[/tex]
Therefore, the sum of the first 8 terms will be:
