Question:
The geometric sequence ; is defined by the formula: [tex]a_1 =10[/tex] ; [tex]a_i =a_{i-1} * \frac{9}{10}[/tex] Find the sum of the first 75 terms in the sequence .
Answer:
[tex]S_{75} = 99.963001151[/tex]
Step-by-step explanation:
Given
[tex]a_1 =10[/tex]
[tex]a_i =a_{i-1} * \frac{9}{10}[/tex]
Required
Find the sum of 75 terms
Given that the sequence is geometric;
First, the common ratio has to be calculated;
The common ratio is defined as follows;
[tex]r = \frac{a_{i}}{a_{i-1}}[/tex]
Let [tex]i = 2[/tex]
[tex]r = \frac{a_{2}}{a_{2-1}}[/tex]
[tex]r = \frac{a_{2}}{a_{1}}[/tex]
So,
[tex]a_i =a_{i-1} * \frac{9}{10}[/tex] becomes
[tex]a_2 =a_{2-1} * \frac{9}{10}[/tex]
[tex]a_2 =a_{1} * \frac{9}{10}[/tex]
Divide through by [tex]a_1[/tex]
[tex]\frac{a_2}{a_1} =\frac{a_{1} * \frac{9}{10}}{a_1}[/tex]
[tex]\frac{a_2}{a_1} = \frac{9}{10}[/tex]
Recall that [tex]r = \frac{a_{2}}{a_{1}}[/tex]
So, [tex]r = \frac{9}{10}[/tex]
Given that r < 1;
The sum of n terms is calculated as thus;
[tex]S_n = \frac{a(1-r^n)}{1-r}[/tex]
To calculate the sum of the first 75 terms, we have the following parameters
[tex]n = 75\\a = a_1 = 10\\r = \frac{9}{10} = 0.9[/tex]
[tex]S_n = \frac{a(1-r^n)}{1-r}[/tex] becomes
[tex]S_{75} = \frac{10(1-0.9^{75})}{1-0.9}[/tex]
[tex]S_{75} = \frac{10(1-0.9^{75})}{0.1}[/tex]
[tex]S_{75} = 100(1-0.9^{75})[/tex]
[tex]S_{75} = 100(1-0.00036998848)[/tex]
[tex]S_{75} = 100(0.99963001151)[/tex]
[tex]S_{75} = 99.963001151[/tex]
