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My son is doing math homework with exponents. The problem is: "If Joe has 1 tribble on Monday and every tribble splits into 3 tribbles each night, how many tribbles on Friday morning?"

So, $ 3^0+3^1+3^2+3^3+3^4 $ is the answer (I think!), but my question:

Is there some generalized form of $ \sum^{i}_0 b^i $ ? I looked at these possible answers but didn't understand, so if you can elaborate it would be appreciated:

KJ7LNW
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  • You answer isn't correct. $1$ tribble splits into $3$, so on Monday he has $1$, Tuesday he has $3$, Wednesday he has $9=3^2$, and so on. So by Friday he would have $3^4=81$ tribbles. – Bonnaduck Oct 22 '21 at 05:31

1 Answers1

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This is the geometric series.

Assume an arbitraty base $q\neq 1$ then

$$S(n)= 1+ q+ q^2+\ldots+q^n$$ $$qS(n)=q+q^2+\ldots+q^{n+1}$$

Subtract both equations and see what happens.

Edit: Subtraction yields

$$(1-q)S(n)=1-q^{n+1} \implies S(n)=\dfrac{1-q^{n+1}}{1-q}$$

MachineLearner
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  • So you multiplied q on both sides and then subtracted? Is this a common technique for reducing geometric series? (Its been since like 1996 when I did those; amazing what kids are able to remind us of!) Looks like the answer would be, q^(n+1)-1, correct? – KJ7LNW Oct 21 '21 at 20:57
  • I edited my answer. – MachineLearner Oct 21 '21 at 21:00
  • Of course it is likely that K7's son should not do it this way. He should compute the number on each day. – GEdgar Oct 21 '21 at 21:04
  • @GEdgar: its true, but I was curious! I can show him the answer and try to explain how the closed form works, but really his class is introducing concepts of exponentiation. – KJ7LNW Oct 21 '21 at 21:27