Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2007-09-29 08:55:08

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Geometric Series

.

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

Offline

#2 2007-09-29 10:21:32

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.

Offline

#3 2007-09-29 11:46:26

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

Offline

#4 2007-09-29 12:24:06

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. wink

Offline

#5 2007-09-30 08:50:59

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. tongue
­

Offline

#6 2007-10-01 11:13:56

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.)

Offline

Board footer

Powered by FluxBB