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: $a_1 =10$ ; $a_i =a_{i-1} * \frac{9}{10}$ Find the sum of the first 75 terms in the sequence .

Answer:

$S_{75} = 99.963001151$

Step-by-step explanation:

Given

$a_1 =10$

$a_i =a_{i-1} * \frac{9}{10}$

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;

$r = \frac{a_{i}}{a_{i-1}}$

Let $i = 2$

$r = \frac{a_{2}}{a_{2-1}}$

$r = \frac{a_{2}}{a_{1}}$

So,

$a_i =a_{i-1} * \frac{9}{10}$ becomes

$a_2 =a_{2-1} * \frac{9}{10}$

$a_2 =a_{1} * \frac{9}{10}$

Divide through by $a_1$

$\frac{a_2}{a_1} =\frac{a_{1} * \frac{9}{10}}{a_1}$

$\frac{a_2}{a_1} = \frac{9}{10}$

Recall that $r = \frac{a_{2}}{a_{1}}$

So, $r = \frac{9}{10}$

Given that r < 1;

The sum of n terms is calculated as thus;

$S_n = \frac{a(1-r^n)}{1-r}$

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

$n = 75\\a = a_1 = 10\\r = \frac{9}{10} = 0.9$

$S_n = \frac{a(1-r^n)}{1-r}$ becomes

$S_{75} = \frac{10(1-0.9^{75})}{1-0.9}$

$S_{75} = \frac{10(1-0.9^{75})}{0.1}$

$S_{75} = 100(1-0.9^{75})$

$S_{75} = 100(1-0.00036998848)$

$S_{75} = 100(0.99963001151)$

$S_{75} = 99.963001151$