Find the partial sum S_n for a geometric series such that a_{4} = 216, a_{9} = 52488, and n = 10.
I assume that a_{n} ios the nth term of the series.
The ratio of successive terms is 3, since
(52488/216)^(1/5) = 3
a_{n} = (216/81)* 3^n
Add up the first 10 terms for the partial sum S_n
To find the sum of a geometric series, we can use the formula:
S_n = a * (1 - r^n) / (1 - r)
Where:
- S_n is the sum of the series up to the nth term,
- a is the first term of the series,
- r is the common ratio of the series,
- n is the number of terms in the series.
Given the values a_4 = 216, a_9 = 52488, and n = 10, we need to find the first term (a) and the common ratio (r) before we can substitute them into the formula.
Step 1: Find the common ratio (r)
To find the common ratio (r), we can divide any term by its previous term:
r = a_9 / a_4 = 52488 / 216
Step 2: Find the first term (a)
We know that a_4 = 216, but we need to find the first term, a. To do so, we can use the formula:
a = a_4 / (r^4)
Substituting the given values:
a = 216 / (r^4)
Step 3: Calculate the sum (S_n)
Now that we have the first term (a) and the common ratio (r), we can substitute these values into the formula for the sum of a geometric series:
S_n = a * (1 - r^n) / (1 - r)
Substituting the given values:
S_10 = a * (1 - r^10) / (1 - r)
With a and r calculated, we can substitute their values into the formula to find S_10.