C2 Geometric Series

The sum of the geometric series 1+r+r²+... is k times the sum of the series 1-r+r²..., where k>0. Express r in terms of k.

By equating the sums of both series, I've got (1-rⁿ)/(1-r) = k(1-(-r)ⁿ)/(1+r), but I'm struggling from here. Help appreciated.

Reply 1

BabyMaths

I think you should be looking at the sum to infinity.

Reply 2

Julii92

OP

Original post by BabyMaths

I think you should be looking at the sum to infinity.

The question doesn't say anything about the value of r - would you think I could assume that -1<r<1?
Thanks

Reply 3

TrueGrit

If |r| > 1 then the sum of the series is infinity, so presumable it means not.

Also are you sure about the second series: 1-r+r2 the 'minus' ?

(edited 11 years ago)

Reply 4

Imposition

Original post by TrueGrit

If |r| > 1 then the sum of the series is infinity, so presumable it means not.

Also are you sure about the second series: 1-r+r2 the 'minus' ?

Might mean you multiply by

-r

Reply 5

TrueGrit

Original post by Imposition

Might mean you multiply by

-r

I don't know I didn't write the question...

Reply 6

aznkid66

1

Original post by TrueGrit

I don't know I didn't write the question...

More like the question most likely means a geometric series with a ratio of -r. If the minus were a plus, there wouldn't be much of a question.

Reply 7

Imposition

Original post by TrueGrit

I don't know I didn't write the question...

I meant that the common ratio is

-r

I think the OP's missing a part of the question out, sum to infinity maybe.

Reply 8

Julii92

OP

Original post by Imposition

I meant that the common ratio is

-r

I think the OP's missing a part of the question out, sum to infinity maybe.

Just re-checked the question, it's actually written "the sum of the infinite geometric series 1+r+..." but I'm not sure what it means by infinte series.

Reply 9

BabyMaths

So, it took you 13 hours to get around to reading the question properly.

:spank:

If |r|<1 then as you add terms the sum approaches some particular value. For example S = 1+1/2+1/4+1/8+1/16......

The sums are

1
1+1/2=3/2
1+1/2+1/4=7/4
1+1/2+1/4+1/8=15/8

and it's clearly approaching 2 and you can get as close to 2 as you like.

(edited 11 years ago)

Reply 10

ztibor

10

Original post by Julii92

The sum of the geometric series 1+r+r²+... is k times the sum of the series 1-r+r²..., where k>0. Express r in terms of k.

By equating the sums of both series, I've got (1-rⁿ)/(1-r) = k(1-(-r)ⁿ)/(1+r), but I'm struggling from here. Help appreciated.