Given the geometric Series:
1/5 + 1/25 + 1/125 + .........
(a) Find the next two terms.
(b) Determine if it is possible to find s∞.
(c) If possible, find s∞ and then express this sum using the sigma notation.
Any help guys? I can understand that the next term will be 1/625 and then 1/3125? but I can't understand b and c.
A geometric sequence a sequence of the form:
a_n = a*r^(n - 1) for n = 1,2,3,4,5... where a_n is the nth term, a is the first term, r is the common ratio, and n is the current number of the term.
here we have a = ar^0 = 1/5
and we have a_2 = ar = 1/25
=> a_2/a = ar/ar^0 = r = (1/25)/(1/5) = 1/5
so our sequence is: a_n = (1/5)(1/5)^(n - 1) for n = 1,2,3,4,5...
so the next two terms are a_4 and a_5
a_4 = (1/5)(1/5)^(4 - 1) = (1/5)^4 = 1/625
a_5 = (1/5)(1/5)^(5 - 1) = (1/5)^5 = 1/3125
any questions?
(b) Determine if it is possible to find s∞.
since |r|<1 it is possible to find S_infinity
For a geometric sequence where |r|<1, the sum to infinity of all the terms is given by:
(c) If possible, find s∞ and then express this sum using the sigma notation.
Any help guys? I can understand that the next term will be 1/625 and then 1/3125? but I can't understand b and c.
S_infinity = a/(1 - r) = (1/5)/(1 - (1/5)) = (1/5)/(4/5) = 1/4
express this in sigma notation we have:
SUM{n=0 to infinity}ar^n = SUM{n=0 to infinity}(1/5)(1/5)^n = SUM{n=0 to infinity}(1/5)^(n+1)
the sum should mean the sumation sign, write n = 0 at the bottom and the infinity symbol at the top