Determine the sum of the first seven terms of the geometric series in which ...

t8 = t5 * r^3, so

r^3 = t8/t5 = -40/5 = -8
so, r = -2

t5 = ar^4 = 16a, so a = 5/16

S7 = a(1-r^7)/(1-r)
= 5/16 (1+2^7)/(1+2) = 5/16 * 129/3 = 215/16 = 13.4375

Can you get r by plugging in numbers into an equation, i.e. ar^n-1 and then solving by substitution/elimination?

That was my initial thought but I couldn't figure out how...

Not my orinigal post but the one right above this one, can someone answer?

well, that's basically what I did.

t5 = ar^4
t8 = ar^7
r^3 = t8/t5

a = t5/r^4 = t5 / ∛(t8/t5)^4 = t5 ∛(t5/t8)^4

S7 = t5 ∛(t5/t8)^4 (1-(t8/t5)^(7/3))/(1-∛(t8/t5))

Now just plug in the numbers and let 'er rip!

Thanks again Steve!

To determine the sum of the first seven terms of a geometric series, we need to find the common ratio and the first term of the series. We can use the given information and a system of equations to find these values.

Let's start by using the formula for the nth term of a geometric series:

tn = a * r^(n-1)

Where "tn" is the nth term, "a" is the first term, "r" is the common ratio, and "n" is the term number.

We are given two equations: t5 = 5 and t8 = -40.

Using the formula above, we can write these equations as:

5 = a * r^(5-1) [equation 1]
-40 = a * r^(8-1) [equation 2]

To solve this system of equations, we can divide equation 2 by equation 1 to eliminate "a":

(-40)/(5) = (a * r^(8-1))/(a * r^(5-1))

Simplifying, we get:

-8 = r^3

Now, we need to find the value of "r" to determine the common ratio. We can take the cube root of both sides of the equation:

r = ^(3√)(-8)

The cube root of -8 is -2, so:

r = -2

Now that we have the value of "r", we can substitute it back into equation 1 to find the value of "a".

5 = a * (-2)^(5-1)

Simplifying, we get:

5 = a * (-2)^4
5 = a * 16
a = 5/16

So, the first term of the series is a = 5/16 and the common ratio is r = -2.

Now, we can use the formula for the sum of a geometric series:

Sn = a * (1 - r^n) / (1 - r)

Where "Sn" is the sum of the first n terms of the series.

We want to find the sum of the first seven terms, so n = 7.

Using the values we found, we can substitute them into the formula:

S7 = (5/16) * (1 - (-2)^7) / (1 - (-2))

Calculating this expression, we get:

S7 = (5/16) * (1 - 128) / 3
S7 = (5/16) * (-127) / 3
S7 = -635/48

Therefore, the sum of the first seven terms of the given geometric series is -635/48.