1. ## Problem (Geometric Progressions)

Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week, 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher's forecasts for

(i) The number of copies that will be sold in the 20th week after publication.
(ii) The total number of copies sold during the first 20 weeks after publication.
(iii) The total number of copies that will ever be sold.

2. Originally Posted by Lonehwolf
Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week, 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher's forecasts for

(i) The number of copies that will be sold in the 20th week after publication.
(ii) The total number of copies sold during the first 20 weeks after publication.
(iii) The total number of copies that will ever be sold.
First term ,a = 3000

Common ratio,

$r = \frac{2400}{3000} = \frac{4}{5}$

i) nth term of the sequence is given by

$
T_n =a\times r^{n-1}
$

$
T_{20} =3000\times (\frac{4}{5})^{20-1}
$

ii) Sum upto nth term of GP is given by

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

Try putting the values yourself

iii) Sum upto infinite terms of a GP with

$|r|< 1$

$= \frac{a}{1-r}$

Put the values