Geometric progression

**mattty** · January 21st 2009, 08:29 AM

Each stroke of a pump removes 8.2% of the remaining air from a container. What percent of the air remains after 50 strokes?

I originally thought this was simple, use the formula $a_n=a_1r^{n-1}$
where
$a_1=1$
$r=0.918$
and $n=50$

substituting in these values i get
$a_{50}=1*0.918 ^{50-1}$
$a_{50}=0.015111$
and therefore ~1.51%
however, the solution in my textbook shows the intermediary step to be
$a_{50}=1*0.918 ^{50}$
and final solution as
$a_{50}=0.01387$

Which brings me to my main question: Could somebody explain to me why the textbook's solution shows $0.918^{50}$ instead of $0.918^{50-1}$ ? Is it something to do with $a_{1}=1$ ? a mistake in the book? or something else?

thanks in advance

**Soroban** · January 21st 2009, 08:51 AM

Hello, mattty!

You miscounted . . .

Each stroke of a pump removes 8.2% of the remaining air from a container.
What percent of the air remains after 50 strokes?

I originally thought this was simple, use the formula $a_n\:=\:a_1r^{n-1}$ . . . . no
where: . $a_1=1,\;r=0.918,\;n=50$

You should be using: . $a_n \:=\:a_1r^{{\color{red}n}}$

Check out the initial condition.

When $n = 0,\;a_o \:=\:a_1r^0 \quad\Rightarrow\quad a_o = a_1$

When no air has been pumped out, the amount is the initial amount, $a_1$

**mattty** · January 21st 2009, 09:05 AM

that's a forehead smack moment if i ever saw one...
yeah, i can see now, when $n=1$ 100% of the air has been pumped out
thanks for pointing out the obvious.... i need that kind of help a lot -_-

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