Question

A manufacturing company that has just
located in a small community will pay two million dollars
per year in salaries. It has been estimated that 60% of these
salaries will be spent in the local area, and 60% of the
money spent will again change hands within the community.
This process, called the multiplier effect, will be repeated
ad infinitum. Find the total amount of local spending
that will be generated by company salaries.

Answer 1

Do you know the formula for the sum of a geometric sequence? What is the limit as the number of terms goes to infinity?

Answer 2

no, could you show me

Answer 3

The sum of a geometric sequence is \[\sum _{k=0}^{n-1} a r^k = \frac{a \left(1-r^n\right)}{1-r}\]

What happens to that as \(n\rightarrow\infty\)? (Remember that \(|r| < 1\) if this converges!)

Answer 4

It seems like converges cause the numerator becomes negative, and each tiem more larger

Answer 5

oh man but I'm not good at this at all. .. so what is the next step ?

Answer 6

in fact i don't know what is a and r here ( in the problem)

Answer 7

As \(n\rightarrow\infty\), that \(r^n\) term goes to 0, leaving us with just \(\large a*\frac{1}{1-r}\). \(a\) is the amount of money invested...

Answer 8

so you mean\[\lim_{n \rightarrow \infty} r ^{n} = 0\] cause r ^ -10 ^6 and so on is more near to zero

Answer 9

ok understood, but how is this related with the problem ?

Answer 10

I guess now i have to plug n chuff values on the equation left

Answer 11

\[\sum_{0}^{\infty}=\frac{ 60 }{ 1-60 }\] right ?

Answer 12

sorry, juggling too many questions at once!

In a geometric sequence, we have terms \({a, a*r, a*r*r, a*r*r*r, ... a*r^n}\)

For our application, \(a = \$2,000,000\) and \(r = 60\% = 0.60\)

So the first round of circulation, $2 million is spent. The second round, $2 million *0.6 = $1.2 million is spent. The third round, $1.2 million * 0.6 = $0.72 million, and so on.

Answer 13

We can factor out that value of \(a\):

\[\sum _{k=0}^{n-1} a r^k = a \sum_{k=0}^{n-1} r^k = a\frac{ \left(1-r^n\right)}{1-r}\]
As \(n\rightarrow\infty\), the fraction approaches \(\large\frac{1}{1-r}\) so the total money spent and spent and spent is just \[a\frac{1}{1-r} = \$2000000(\frac{1}{1-0.60}) = \]

You can test this for reasonability at the other end by saying that none of the money is respent by setting \(r = 0\). If you do that, what do you get?

Answer 14

oh i see it behaves like a linear operator

Answer 15

I'll have to take your word for that :-)

Here's a plot of the effect as \(r\) goes from \(0\rightarrow 1\)

Clearly, if you can boost that percentage of money recirculating in the local community even just a little bit, it has a bit impact!

Answer 16

big, not bit

Answer 17

awesome you know a lot of math

Answer 18

Thanks bro

Answer 19

You're welcome!