Thread: Finding the next terms in series

1. Finding the next terms in series

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
sand 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.

2. Originally Posted by bobchiba
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
sand 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.

b) Only possible if -1 < r < 1. In this case r = 1/5 so yes (r is common ratio)

3. Originally Posted by bobchiba
Given the geometric Series:
1/5 + 1/25 + 1/125 + .........

(a) Find the next two terms.

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

(c) If possible, find
sand 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.

For a geometric sequence where |r|<1, the sum to infinity of all the terms is given by:

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