C2 Series Help Please

A geometric series has first term a and common ratio r where |r| > 1. The sum of the first n terms of the series is denoted by Sn.

Given that S4 = 10 x S2.

a) Find the value of r.

I've tried finding the formula for the sum of the first four terms and equating that to ten times the sum of the first two terms so im then left with two unknowns (r and a) and i thought that i'd be able to solve it, but i cant because of the powers on some of the r's.

Could someone please explain how this can be done?

Reply 1

DeanK2

Where is your working - (you also mean [r] < 1).

Reply 2

electriic_ink

15

iali4

and i thought that i'd be able to solve it, but i cant because of the powers on some of the r's.

Please expand on this.

Reply 3

iali4

OP

As in the formula for geometric series is = a(r^n -1) over (r-1) so there are going to be r's to the power of 4 and 2, cos thats what n is, or am i totally wrong?

And to DeanK2 i double checked and on the sheet it says |r| > 1

Reply 4

Tla

8

|r| > 1

is strange, but if you're sure...

r < -1

or

r > 1

.

\frac{a(1-r^{4})}{1-r} = \frac{10(a(1-r^{2}))}{1-r}

a(1-r^{4}) = 10(a(1-r^{2}))

1-r^{4} = 10(1-r^{2})

1-r^{4} = 10-10r^{2}

Rearrange this to get...

r^{4}-10r^{2}+9 = 0

This is a disguised quadratic equation, you should be able to solve it. Only use solutions where

|r| >1

in your final answer.

Reply 5

electriic_ink

15

iali4

As in the formula for geometric series is = a(r^n -1) over (r-1) so there are going to be r's to the power of 4 and 2, cos thats what n is, or am i totally wrong?

And to DeanK2 i double checked and on the sheet it says |r| > 1

Here's what I got.

a-ar^4 = 10(a - ar^2) (Because S₄ = 10S₂ and the "1-r"s cancel)
1 - r^4 = 10 - 10r^2 (The a terms cancel)
r^4 - 10r^2 = -9 (Rearrange)
Let b=r^2
b^2-10b+9=0 (Substitute in "b" and rearrange)
Therefore, b=1 and b=9
r=1, r=3, r=-1, r=-3
r=3 because r=1, r=-1, r=-3 are not within the range.

Reply 6

iali4

OP

Yeah i got that far too and then cross multiplied the (r-1) , but my teacher said that if |r| > 1 then to use the other formula a(r^n -1) over (r-1). so then i was left with ar^4 - ar + a = 10ar^3 - 10ar + 10a. but i cant see how to work it out from that

Reply 7

iali4

OP

electriic_ink

Here's what I got.

a-ar^4 = 10(a - ar^2) (Because S₄ = 10S₂ and the "1-r"s cancel)
1 - r^4 = 10 - 10r^2 (The a terms cancel)
r^4 - 10r^2 = -9 (Rearrange)
Let b=r^2
b^2-10b+9=0 (Substitute in "b" and rearrange)
Therefore, b=1 and b=9
r=1, r=3, r=-1, r=-3
r=3 because r=1, r=-1, r=-3 are not within the range.