Question

The geometric sequence ; is defined by the formula: a i a 1 =10 a i =a i-1 * 9/10 Find the sum of the first 75 terms in the sequence .

Answer 1

Question:

The geometric sequence ; is defined by the formula:
`a_1 =10` ;
`a_i =a_(i-1) * (9)/(10)` Find the sum of the first 75 terms in the sequence .

Answer:

`S_(75) = 99.963001151`

Explanation:

Given

`a_1 =10`

`a_i =a_(i-1) * (9)/(10)`

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;

`r = (a_(i))/(a_(i-1))`

Let
`i = 2`

`r = (a_(2))/(a_(2-1))`

`r = (a_(2))/(a_(1))`

So,

`a_i =a_(i-1) * (9)/(10)` becomes

`a_2 =a_(2-1) * (9)/(10)`

`a_2 =a_(1) * (9)/(10)`

Divide through by
`a_1`

`(a_2)/(a_1) =(a_(1) * (9)/(10))/(a_1)`

`(a_2)/(a_1) = (9)/(10)`

Recall that
`r = (a_(2))/(a_(1))`

So,
`r = (9)/(10)`

Given that r < 1;

The sum of n terms is calculated as thus;

`S_n = (a(1-r^n))/(1-r)`

To calculate the sum of the first 75 terms, we have the following parameters

`n = 75\\a = a_1 = 10\\r = (9)/(10) = 0.9`

`S_n = (a(1-r^n))/(1-r)` becomes

`S_(75) = (10(1-0.9^(75)))/(1-0.9)`

`S_(75) = (10(1-0.9^(75)))/(0.1)`

`S_(75) = 100(1-0.9^(75))`

`S_(75) = 100(1-0.00036998848)`

`S_(75) = 100(0.99963001151)`

`S_(75) = 99.963001151`