Find the sum of the infinite geometric series, if possible.
12 + 10 + 25/3 + 125/18 + ...
a = 12
r = 5/6
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html
a [ 1/(1-r) ] = 12 [ 1 / (1/6) = 12 * 6
To determine if the given series has a sum, we can check if the common ratio is between -1 and 1.
The common ratio (r) of a geometric series is found by dividing any term by the previous term.
In this case, let's take the second term divided by the first term:
10/12 = 5/6.
Since the absolute value of the common ratio (|r| = |5/6|) is less than 1, we can conclude that the given series has a sum.
To find the sum (S) of an infinite geometric series, you can use the formula:
S = a / (1 - r),
where 'a' represents the first term and 'r' is the common ratio.
In this case:
a = 12,
r = 5/6.
Now substitute these values into the formula to find the sum:
S = 12 / (1 - 5/6).
To simplify this, we need to find a common denominator:
1 - 5/6 = 6/6 - 5/6 = 1/6.
So, the sum of the infinite geometric series is:
S = 12 / (1/6) = 12 * (6/1) = 72.
Therefore, the sum of the given infinite geometric series is 72.
To find the sum of an infinite geometric series, we need to determine whether the series is convergent or divergent.
A geometric series is convergent if the common ratio (r) is between -1 and 1 (excluding -1 and 1). In this case, the common ratio can be found by dividing any term by its preceding term.
Let's calculate the common ratio (r):
r = (25/3) / 10 = 5/6
Since the common ratio (5/6) is between -1 and 1, the series is convergent.
To find the sum of a convergent geometric series, we can use the formula:
S = a / (1 - r)
Where:
- S represents the sum of the series
- a is the first term of the series
- r is the common ratio
In this case:
a = 12 (the first term)
r = 5/6 (the common ratio)
Plugging in the values:
S = 12 / (1 - 5/6)
Now, simplify the expression within the parentheses:
S = 12 / (1/6)
Next, divide to get the final result:
S = 72
Therefore, the sum of the given infinite geometric series is 72.