We have: #<6.75, - 4.5, 3, ...>#
This is a geometric sequence with a common ratio #r = - frac(4.5)(6.75) = - frac(3)(4.5) = - frac(2)(3)#.
The formula for finding the sum of an infinite geometric sequence is #S_(infty) = frac(u_(1))(1 - r)#; #|r| < 1#.
For our sequence, the first term #u_(1)# is #6.75#, and the common ratio is #- frac(2)(3)#:
#Rightarrow S_(infty) = frac(6.75)(1 - (- frac(2)(3)))#
#Rightarrow S_(infty) = frac(6.75)(1 + frac(2)(3))#
#Rightarrow S_(infty) = frac(6.75)(frac(4)(3))#
#Rightarrow S_(infty) = frac(frac(675)(100))(frac(4)(3))#
#Rightarrow S_(infty) = frac(675)(100) cdot frac(3)(4)#
#Rightarrow S_(infty) = frac(2025)(400)#
#therefore S_(infty) = 5.0625#
Therefore, the sum of all the terms in the sequence is #5.0625#.
