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Question 235791: Hi! My question is: Find the partial sum S_n for a geometric series such that a_{4} = 216, a_{9} = 52488, and n = 10. I think I know the formula to use (Sn = a1 (1-r^n) / (1-r)) but 1.) I'm not completely sure and @. I don't know what to plug in where. Thank you very much!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Your basic formula is correct as shown here !!!!!
In a geometric series, each succeeding term is equal to the last term multiplied by a common ratio.
In your geometric series, you are given that:
a[4] = 216
a[9] = 52488
You need to find a[1] and r.
In general,
a[n] = a[1] * r^(n-1)
and:
a[n] = r*a[n-1]
This second equation can be generalized to
a[n] = r^(n-x) * a[x]
where:
n is the nth term in the series.
x is the xth term in the series.
In your problem, you have:
a[9] = 52488
a[4] = 216
This means that, using the generalized formula, we can solve for r as follows:
n = 9
x = 4
n-x = 9-4 = 5
a[9] = 52488
a[4] = 216
The generalized formula becomes:
52488 = r^5 * 216
Divide both sides of this equation by 216 to get:
52488/216 = r^5
Simplify to get:
243 = r^5
Take the 5th root of both sides to get:
r = 3
Now you want to solve for a[1] which is the first term in your sequence.
We have:
a[1] = ?????
a[4] = 216
n = 4
x = 1
n-x = 3
Using the generalized formula again, we get:
216 = a[1] * r^3
Solve for a[1] to get:
a[1] = 216 / 3^3 = 8
Your first term in this geometric series is:
a[1] = 8
Your ratio is:
r = 3
The number of terms in your geometric series is:
n = 10
The sum of a geometric series is given by the equation:
S[n] = a[1] * (1-r^n)/(1-r)
The sum of this geometric series becomes:
S[10] = 8 * ((1-3^10)/(1-3)
because:
a[1] = 8
r = 3
n = 10
This makes:
S[10] = 8 * ((-59048)/(-2))
Which becomes:
S[10] = 236192
That's your answer.
Some additional information regarding geometric sequences can be found here !!!!! and here!!!!!
FYI,
Since a[1] = 8, we can find a[4] and a[9] using the formula for the nth term in a geometric sequence.
That formula is:
a[n] = a[1] * r^(n-1)
a[4] = 8 * 3^(3) = 8 * 27 = 216
a[9] = 8 * 3^(8) = 8 * 6561 = 52488
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