how can I calculate $r$ in $r + r^2 + r^3 + r^4 = 100$? [closed]

As the title says, I want to know how to solve this equation. Just cannot find a way

I want to know how much % profit someone has to make EVERY MONTH in average from a stock within $4$ months, to double the money. In other words: I have to make r % profit on every single month of the 4 month to double (r=100%) the money.

• What is $r^5 -1$ ?
– user312097
Oct 11, 2017 at 15:08
• Anyway you need to solve a polynomial that has degree more than $2$.
– user312097
Oct 11, 2017 at 15:11
• Clearly, $2<r<3$. So $r\simeq 2.84922$. Oct 11, 2017 at 15:12
• thanks guys. for me as a math noob, can you show me the exact way? I have also changed the description of my question to what my goal is with this equation. Oct 11, 2017 at 15:15
• You can factor, but then to find the actual solutions you must use newtons method Oct 11, 2017 at 15:17

The underlying problem is that you've got the wrong equation.

If you make a profit of $r$% after each month, and started with $n$, then after one month, you'd have $n\left(1+\frac{r}{100}\right)$, after two months, $n\left(1+\frac{r}{100}\right)^2$, and in general, after $m$ months:

$$n\left(1+\frac{r}{100}\right)^m$$

When $m=4,$ you want to have $2n$, so you want:

$$\left(1+\frac{r}{100}\right)^4=2$$

which yields:

$$r=100(\sqrt[4]2-1)\approx 18.92$$

• Thanks Thomas!! This helped me a lot! Oct 11, 2017 at 15:57

Why would you need to know something like this: $$-1/4-1/12\,\sqrt {3}\sqrt {{\frac {4\, \left( 2260+6\,\sqrt {48382278} \right) ^{2/3}-5\,\sqrt [3]{2260+6\,\sqrt {48382278}}-4808}{\sqrt [3] {2260+6\,\sqrt {48382278}}}}}+1/12\,\sqrt {6}\sqrt {{\frac {-2\,\sqrt {{\frac {4\, \left( 2260+6\,\sqrt {48382278} \right) ^{2/3}-5\,\sqrt [ 3]{2260+6\,\sqrt {48382278}}-4808}{\sqrt [3]{2260+6\,\sqrt {48382278}} }}} \left( 2260+6\,\sqrt {48382278} \right) ^{2/3}+15\,\sqrt {3}\sqrt [3]{2260+6\,\sqrt {48382278}}-5\,\sqrt [3]{2260+6\,\sqrt {48382278}} \sqrt {{\frac {4\, \left( 2260+6\,\sqrt {48382278} \right) ^{2/3}-5\, \sqrt [3]{2260+6\,\sqrt {48382278}}-4808}{\sqrt [3]{2260+6\,\sqrt { 48382278}}}}}+2404\,\sqrt {{\frac {4\, \left( 2260+6\,\sqrt {48382278} \right) ^{2/3}-5\,\sqrt [3]{2260+6\,\sqrt {48382278}}-4808}{\sqrt [3] {2260+6\,\sqrt {48382278}}}}}}{\sqrt [3]{2260+6\,\sqrt {48382278}} \sqrt {{\frac {4\, \left( 2260+6\,\sqrt {48382278} \right) ^{2/3}-5\, \sqrt [3]{2260+6\,\sqrt {48382278}}-4808}{\sqrt [3]{2260+6\,\sqrt { 48382278}}}}}}}}$$

when numerical solution will give you $2.849217207$ approximately.

Edit:

But in all seriousness you must factor and then use alternative methods such a Newton Raphson to solve for the zeroes. Otherwise its virtually impossible by just hand

• I lol'd so hard Oct 11, 2017 at 15:17
• Thanks a lot for your reply. So generally, is the way I want to go the right way to solve the problem as I've written in the description? "I want to know how much % profit someone has to make EVERY MONTH in average from a stock within 44 months, to double the money." In other words: I have to make r % profit on every single month of the 4 month to double (r=100%) the money. Oct 11, 2017 at 15:22
• No, you are not going the right way. Because of compounding, the profit after four months is $(1+r)^4-1$, not $r+r^2+r^3+r^4$. To find the value of r that creates 100% profit after 4 months you need to solve $(1+r)^4-1=1$, which gives a value for r of about 19%. Oct 11, 2017 at 15:47