Difference Equations

**Roam** · June 9th 2009, 05:43 PM

Here's my question:

The answer to this probelm is:

$<br /> S_{n} = 0.076 S_{n-1}<br />$

I don't see how to get this answer, here's my attempt:

Equation has to be of the form $S_{n} = k S_{n-1}$ , where k is some constant. I don't understand how they've got k=0.076

Any explanation would be helpful.

**the_doc** · June 10th 2009, 07:10 AM

Clearly this should be:

$S_n = 0.976 \, S_{n-1}$

so either you've mistyped it or the person who set the question mistyped it putting a zero in place of where the 9 should have been!

**Roam** · June 10th 2009, 11:45 AM

Originally Posted by the_doc

Clearly this should be:

$S_n = 0.976 \, S_{n-1}$

so either you've mistyped it or the person who set the question mistyped it putting a zero in place of where the 9 should have been!

Oh, ok, but how did you get 0.976?

**Random Variable** · June 10th 2009, 11:48 AM

Originally Posted by Roam

Oh, ok, but how did you get 0.976?

$S_{n} - S_{n-1} = -0.024S_{n-1}$

**Roam** · June 10th 2009, 01:12 PM

I'm sorry but I can't follow what you've done, could you please explain a little more?

**the_doc** · June 10th 2009, 01:19 PM

Say after $(n-1)$ years the amount is $S_{n-1}$ so then after one more year you lose 2.4 % which is $0.024$ of $S_{n-1}$ so

the amount lost $= 0.024 S_{n-1}$ so then after $n$ years we have the amount $S_n$ given by the amount at the start of the year, $S_{n-1}$ , minus the amount lost so

$S_n = S_{n-1} - 0.024 S_{n-1}$ ,

$= (1-0.024) S_{n-1}$ ,

$= 0.976 S_{n-1}$ .

Any better?

**skeeter** · June 10th 2009, 01:21 PM

... How did you get 0.976?

100% - 2.4% = 97.6%

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