I'm trying to find the sum of this geometric series but I can't find a common ratio. How do I find it?
1+1/5+1/25+...
take the 2nd and divide by the 1st:
(1/5) ÷ 1 = 1/5
should be the same as the 3rd divided by the 2nd:
(1/25) ÷ (1/5)
= (1/25)(5/1) = 1/5
looks like r = 1/5
sum(all terms) = a/(1-r) = 1/(1-1/5)
= 1/(4/5)
= 5/4
To find the common ratio of a geometric series, you need to compare each term to its preceding term. In this case, let's look at the first few terms of the series:
1 + 1/5 + 1/25 + ...
To find the common ratio, you can divide any term by the preceding term. Let's divide the second term by the first term:
(1/5) ÷ 1 = 1/5
As you can see, the common ratio is 1/5. Each term is obtained by multiplying the preceding term by 1/5.
Now let's calculate the sum of this geometric series. The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
where:
S = sum of the series
a = first term of the series
r = common ratio of the series
In our case, a = 1 (the first term) and r = 1/5 (the common ratio). Plug these values into the formula:
S = 1 / (1 - 1/5)
Simplifying the expression:
S = 1 / (4/5) = 5/4
Therefore, the sum of the geometric series 1 + 1/5 + 1/25 + ... is 5/4.