What is the closed form for the infinite geometric sequence ?

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1 Answer
Mar 9, 2018

x_n=3 *(5/2)^(n-1)

Explanation:

Note that the common ratio is
color(white)("XXX")color(blue)(r)=(15/2)/3=(75/4)/(15/2)=(375/8)/(75/4)=color(blue)(5/2)

Given
color(white)("XXX")color(red)(x_1)=color(red)(3)
and with color(blue)(r)=color(blue)(5/2)

color(white)("XXX")x_2=color(red)3 * (color(blue)(5/4))^1

color(white)("XXX")x_3=x_2 * color(blue)r= color(red)3 * (color(blue)(5/4))^2

color(white)("XXX")x_4=x_3 * color(blue)r= color(red)3 * (color(blue)(5/4))^3

from which we can imply the general formula:
color(white)("XXX")x_color(magenta)n=color(red)3 * (color(blue)(5/4))^(color(magenta)n-1)