SOLUTION: 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))

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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