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+...
you can't see the common ratio between 1, 5, 25?
Hmmm.
a = 1
r = 1/5
S = a/(1-r) = 1/(1 - 1/5) = 1/(4/5) = 5/4
Anyway, if you suspect a geometric series, just divide any term by the preceding term to get the common ratio
(1/5) / 1 = 1/5
(1/25) / (1/5) = 1/5
. . .
To find the common ratio of a geometric series, you need to look for a pattern among the terms. In this case, we see that each term is obtained by dividing the previous term by 5. Let's examine it step by step:
1st term: 1
2nd term: 1/5 = 1/1 * (1/5)
3rd term: 1/25 = 1/1 * (1/5)^2
and so on...
From this pattern, we can deduce that the common ratio is 1/5. Each term is obtained by multiplying the previous term by 1/5.
Now that you have identified the common ratio, you can use the formula for the sum of a geometric series to find the sum. The formula is as follows:
Sum = a * (1 - r^n) / (1 - r)
Where:
- a represents the first term of the series
- r represents the common ratio
- n represents the number of terms in the series
For this particular geometric series, the first term (a) is 1 and the common ratio (r) is 1/5. The number of terms (n) can be infinite (if we are considering an infinite geometric series) or a specific finite number (in case of a finite geometric series).
If you are looking for the sum of an infinite geometric series, substituting these values into the formula would give you:
Sum = 1 * (1 - (1/5)^∞) / (1 - 1/5)
Note that when the common ratio (r) is between -1 and 1 (inclusive), the infinite geometric series converges and has a finite sum.
However, if you are looking for the sum of a finite geometric series (i.e., considering only a specific number of terms), you would need to substitute the value for 'n' into the formula.