Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Geometric Series

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Last edited by Daniel123 (2007-10-03 07:08:14)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Geometric Series

Let a[sub]n[/sub] be the amount to be paid at the end of the nth year. Then

This is a recurrence relation of the form

the solution of which is

Hence, if N years are required to pay off the mortgage, you need a[sub]N+1[/sub] ≤ 0, i.e.

If my working is correct, that should give N ≥ 24.36. So 25 years are required to pay everything off.

Hunt

Member

Registered: 2007-09-29

Posts: 9

Re: Geometric Series

Hello JaneFairFax , Can you please give a link or show how the following was derived :

thanks

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Geometric Series

I tried the sequence for the first few values of n, got a formula and verified it by mathematical induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Geometric Series

Right. We have

If you carry on, you should be able to convince yourself that

Now

is a sum of a geometric series. The formula is

So you have a formula for a[sub]n[/sub]. So use it.
­

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Geometric Series

If your post has [math] tags in it, it won’t be deleted. To delete it, edit your post and get rid of the [math] tags first. (Then you can delete it.)