Question

Please help! I've been stuck on these questions for weeks now!

Share 5 cookies with 4 people. How much cookie should each person get? Express your answer as a series.

Share 2 cookies with 3 people. How much cookie should each person get? Express your answer as a series.

Answer 1

What series do they want? I thought series is the sum of a sequence.

Answer 2

The first is a finite geometric series. The second is an infinite series. I thought the same thing. That's why I've been so stuck on this

Answer 3

So I have to express \(\dfrac{5}{4}\) as a infinite geometric series?

Answer 4

Finite

Answer 5

\[
\begin{align*}
r+r^2&=\frac{5}{4}\\
r^2+r-\frac{5}{4}&=0\\
r&=\frac{1}{2}(\sqrt{6}-1)
\end{align*}
\]
The geometric sequence is \(\left(\frac{1}{2}(\sqrt{6}-1)\right)^n\) and the sum of first two terms is \(\dfrac{5}{4}\).

Answer 6

Just create a geometric series that works. You could have at least one geometric series with n-terms which sums to \(\dfrac{5}{4}\) for every n I think.

Answer 7

I'm sorry, I'm still confused..

Answer 8

I think they explained enough.

Answer 9

The sum of an infinite geometric series is (see https://en.wikipedia.org/wiki/Geometric_series#Formula)
is
\[S=a+ar+ar^2+ar^3+...=\sum_{k=0}^{\infty}ar^k= \frac{ a }{ 1-r }\]

we want this to add up to 5/4
Let's set a=1 (we can choose any number except 0, so keep it simple)
we now want the sum to be equal to 5/4 so it must be that
\[ \frac{ 1 }{ 1-r }= \frac{5}{4}\]
now solve for r. I would cross-multiply
\[ 4 = 5(1-r) \\ 4= 5 -5r\]
add -4 to both sides
\[ 0 = 5 - 4 - 5r \\ 0 = 1 - 5r\]
add +5r to both sides
\[ 5r= 1 \]
divide both sides by 5
\[ r= \frac{1}{5} \]

that means the infinite series with a=1 and r= 1/5 is:
\[ 1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + ... \]