Find k so f(n^2 + k) = f(n)Xf(n + 1) for f(n) = n^2 - n + 2

[TangoFoxtrotGolf]

Let f(n) = n[sup:2bpiakor]2[/sup:2bpiakor] - n + 2

Find a value for k such that the equation

f(n[sup:2bpiakor]2[/sup:2bpiakor] + k) = f(n) X f(n + 1)

holds for all values of n.

f(n[sup:2bpiakor]2[/sup:2bpiakor] + k) = (n[sup:2bpiakor]2[/sup:2bpiakor] + k)[sup:2bpiakor]2[/sup:2bpiakor] - (n[sup:2bpiakor]2[/sup:2bpiakor] + k) + 2
f(n[sup:2bpiakor]2[/sup:2bpiakor] + k) = (n[sup:2bpiakor]4[/sup:2bpiakor] + 2kn[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor]) - (n[sup:2bpiakor]2[/sup:2bpiakor] + k) + 2
f(n[sup:2bpiakor]2[/sup:2bpiakor] + k) = n[sup:2bpiakor]4[/sup:2bpiakor] + 2kn[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - n[sup:2bpiakor]2[/sup:2bpiakor] - k + 2
f(n[sup:2bpiakor]2[/sup:2bpiakor] + k) = n[sup:2bpiakor]4[/sup:2bpiakor] + 2kn[sup:2bpiakor]2[/sup:2bpiakor] - n[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2

f(n + 1) = (n + 1)[sup:2bpiakor]2[/sup:2bpiakor] - (n + 1) + 2
f(n + 1) = (n[sup:2bpiakor]2[/sup:2bpiakor] + 2n + 1) - (n + 1) + 2
f(n + 1) = n[sup:2bpiakor]2[/sup:2bpiakor] + 2n + 1 - n - 1 + 2
f(n + 1) = n[sup:2bpiakor]2[/sup:2bpiakor] + n + 2

f(n) X f(n + 1) = (n[sup:2bpiakor]2[/sup:2bpiakor] - n + 2)(n[sup:2bpiakor]2[/sup:2bpiakor] + n + 2)
f(n) X f(n + 1) = n[sup:2bpiakor]4[/sup:2bpiakor] + 3n[sup:2bpiakor]2[/sup:2bpiakor] + 4

Therefore,

n[sup:2bpiakor]4[/sup:2bpiakor] + 2kn[sup:2bpiakor]2[/sup:2bpiakor] - n[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2 = n[sup:2bpiakor]4[/sup:2bpiakor] + 3n[sup:2bpiakor]2[/sup:2bpiakor] + 4
n[sup:2bpiakor]4[/sup:2bpiakor] + 2kn[sup:2bpiakor]2[/sup:2bpiakor] - n[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2 - n[sup:2bpiakor]4[/sup:2bpiakor] = n[sup:2bpiakor]4[/sup:2bpiakor] + 3n[sup:2bpiakor]2[/sup:2bpiakor] + 4 - n[sup:2bpiakor]4[/sup:2bpiakor]
2kn[sup:2bpiakor]2[/sup:2bpiakor] - n[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2 = 3n[sup:2bpiakor]2[/sup:2bpiakor] + 4
2kn[sup:2bpiakor]2[/sup:2bpiakor] - n[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2 + n[sup:2bpiakor]2[/sup:2bpiakor] = 3n[sup:2bpiakor]2[/sup:2bpiakor] + 4 + n[sup:2bpiakor]2[/sup:2bpiakor]
2kn[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2 = 4n[sup:2bpiakor]2[/sup:2bpiakor] + 4
2kn[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k + 2 - 4 = 4n[sup:2bpiakor]2[/sup:2bpiakor] + 4 - 4
2kn[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k - 2 = 4n[sup:2bpiakor]2[/sup:2bpiakor]
2kn[sup:2bpiakor]2[/sup:2bpiakor] + k[sup:2bpiakor]2[/sup:2bpiakor] - k - 2 - 2kn[sup:2bpiakor]2[/sup:2bpiakor] = 4n[sup:2bpiakor]2[/sup:2bpiakor] - 2kn[sup:2bpiakor]2[/sup:2bpiakor]
k[sup:2bpiakor]2[/sup:2bpiakor] - k - 2 = 4n[sup:2bpiakor]2[/sup:2bpiakor] - 2kn[sup:2bpiakor]2[/sup:2bpiakor]
k[sup:2bpiakor]2[/sup:2bpiakor] - k - 2 = 2n[sup:2bpiakor]2[/sup:2bpiakor](1 - k)
k[sup:2bpiakor]2[/sup:2bpiakor] - k - 2 = 2n[sup:2bpiakor]2[/sup:2bpiakor](-k + 1)
(k - 2)(k + 1) = 2n[sup:2bpiakor]2[/sup:2bpiakor](-k + 1)

I guess that I hoped to find some factor to divide both sides by to reduce further.

I did make several attempts to multiply both sides by -1 trying to get one of the factors on the left-hand side to be the same as the parenthetic factor on the right-hand side. But, none of those attempts yielded a factor that would divide out.

I don't see where to proceed from here?

Or, if I made a mistake early on?

Any help greatly appreciated.

 

[stapel]

I agree with your results for f(n)*f(n + 1) and f(n^2 + k). I'm not sure what you're doing after that, though...?

As near as I can tell, the best you can do is to find k in terms of n:

. . . . .0 = k[sup:10oqmcux]2[/sup:10oqmcux] + 2n[sup:10oqmcux]2[/sup:10oqmcux]k - 4n[sup:10oqmcux]2[/sup:10oqmcux] - k - 2

. . . . .0 = k[sup:10oqmcux]2[/sup:10oqmcux] + (2n[sup:10oqmcux]2[/sup:10oqmcux] - 1)k + (-4n[sup:10oqmcux]2[/sup:10oqmcux] - 2)

. . . . .0 = k[sup:10oqmcux]2[/sup:10oqmcux] + (2n[sup:10oqmcux]2[/sup:10oqmcux] - 1)k - 2(2n[sup:10oqmcux]2[/sup:10oqmcux] - 1)

. . . . .0 = (k + 2(2n[sup:10oqmcux]2[/sup:10oqmcux] - 1))(k - (2n[sup:10oqmcux]2[/sup:10oqmcux] - 1))

Then:

. . . . .k = -2(2n[sup:10oqmcux]2[/sup:10oqmcux] - 1) or k = 2n[sup:10oqmcux]2[/sup:10oqmcux] - 1

Dunno where you're supposed to go from there, though...

 

[Mrspi]

Let f(n) = n[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n + 2

Find a value for k such that the equation

f(n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) = f(n) X f(n + 1)

holds for all values of n.

f(n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) = (n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k)[sup:3uzxq5l7]2[/sup:3uzxq5l7] - (n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) + 2
f(n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) = (n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7]) - (n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) + 2
f(n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2
f(n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k) = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2

f(n + 1) = (n + 1)[sup:3uzxq5l7]2[/sup:3uzxq5l7] - (n + 1) + 2
f(n + 1) = (n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 2n + 1) - (n + 1) + 2
f(n + 1) = n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 2n + 1 - n - 1 + 2
f(n + 1) = n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + n + 2

f(n) X f(n + 1) = (n[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n + 2)(n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + n + 2)
f(n) X f(n + 1) = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4

Therefore,

n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4
n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 - n[sup:3uzxq5l7]4[/sup:3uzxq5l7] = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4 - n[sup:3uzxq5l7]4[/sup:3uzxq5l7]
2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 = 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4
2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 + n[sup:3uzxq5l7]2[/sup:3uzxq5l7] = 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4 + n[sup:3uzxq5l7]2[/sup:3uzxq5l7]
2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 = 4n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4
2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 - 4 = 4n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4 - 4
2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k - 2 = 4n[sup:3uzxq5l7]2[/sup:3uzxq5l7]
2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k - 2 - 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7] = 4n[sup:3uzxq5l7]2[/sup:3uzxq5l7] - 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7]
k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k - 2 = 4n[sup:3uzxq5l7]2[/sup:3uzxq5l7] - 2kn[sup:3uzxq5l7]2[/sup:3uzxq5l7]
k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k - 2 = 2n[sup:3uzxq5l7]2[/sup:3uzxq5l7](1 - k)
k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k - 2 = 2n[sup:3uzxq5l7]2[/sup:3uzxq5l7](-k + 1)
(k - 2)(k + 1) = 2n[sup:3uzxq5l7]2[/sup:3uzxq5l7](-k + 1)

I guess that I hoped to find some factor to divide both sides by to reduce further.

I did make several attempts to multiply both sides by -1 trying to get one of the factors on the left-hand side to be the same as the parenthetic factor on the right-hand side. But, none of those attempts yielded a factor that would divide out.

I don't see where to proceed from here?

Or, if I made a mistake early on?

Any help greatly appreciated.

In a previous post, I suggested a method often used in solving this type of problem. You've correctly arrived at this point:
[sup:3uzxq5l7][/sup:3uzxq5l7]
n[sup:3uzxq5l7][4[/sup:3uzxq5l7] + 2n[sup:3uzxq5l7]2[/sup:3uzxq5l7]k - n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4

On the left side, there are two terms involving n[sup:3uzxq5l7]2[/sup:3uzxq5l7]: 2n[sup:3uzxq5l7]2[/sup:3uzxq5l7]k - n[sup:3uzxq5l7]2[/sup:3uzxq5l7]. Do you see that we can write that as (2k - 1)*n[sup:3uzxq5l7]2[/sup:3uzxq5l7]? And the last three terms don't involve "n" at all, so I'll group those together.

n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + (2k - 1)*n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + (k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2) = n[sup:3uzxq5l7]4[/sup:3uzxq5l7] + 3n[sup:3uzxq5l7]2[/sup:3uzxq5l7] + 4

The only way the expression on the left (which involves k) can be equal to the expression on the right (which does not involve k) is for the coefficients of the various powers of n to be EQUAL.

The coefficients of n[sup:3uzxq5l7]4[/sup:3uzxq5l7] ARE the same in the expressions on the left and right side.

On the left side, the coefficient of n[sup:3uzxq5l7]2[/sup:3uzxq5l7] is (2k - 1). On the right side, the coefficient of n[sup:3uzxq5l7]2[/sup:3uzxq5l7] is 3. The expression on the left can only be equal to the expression on the right if those coefficients are the SAME. So,

2k - 1 = 3

Solve that for k.

The constant on the left side consists of those terms which do NOT contain "n": k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2
The constant on the right side is 4
In order for the left side to be equal to the right side, the constants must be equal as well. So,

k[sup:3uzxq5l7]2[/sup:3uzxq5l7] - k + 2 = 4

That can be solved for k as well. One of the solutions will be the same as you got in the previous step, but one will be different.

Be sure to check both of your solution values for k to make sure they satisfy the conditions of the problem...it's entirely possible that one solution may not check.

This method of equating coefficients is often used in advanced math, so it is well worth your while to learn how to do it.

 

[soroban]

Hello, TangoFoxtrotGolf!

\(\displaystyle \text{Let }\:f(n) \:= \:n^2 - n + 2\)

\(\displaystyle \text{Find a value for }k\text{ such that the equation:}\)

. . \(\displaystyle f(n^2+ k) \:=\: f(n)\cdotf(n + 1)\;\text{ holds for all values of }n.\)


\(\displaystyle f(n^2+k) \:= \n^2+ k)^2 - (n^2 + k) + 2 \:=\: n^4 + 2kn^2 - n^2 + k^2 - k + 2\)

\(\displaystyle f(n + 1) \:=\: (n + 1)^2 - (n + 1) + 2 \:=\: n^2 + n + 2\)

\(\displaystyle f(n)\cdot f(n + 1) \:= \n^2 - n + 2)(n^2 + n + 2) \:=\: n^4 + 3n^2 + 4\)


\(\displaystyle \text{Therefore: }\:n^4 + 2kn^2 - n^2 + k^2 - k + 2 \:= \:n^4 + 3n^2 + 4\)

At this point, you performed some truly masterful manipulation,
. . trying to get the equation into a useful form.
Sadly, it didn't pay off.


I solved it like Stapel did.
. . And made the same error ... which I corrected.

\(\displaystyle \text{The equation becomes: }\;k^2 + (2n^2-1)k - 2(2n^2 +1) \;=\;0\quad\hdots \text{a quadratic in }k.\)
. . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . ^

\(\displaystyle \text{Quadratic Formula: }\;k \;=\;\frac{-(2n^2-1) \pm\sqrt{(2n^2-1)^2 + 8(2n^2+1)}}{2}\)

. . \(\displaystyle k \;=\;\frac{-(2n^2-1) \pm\sqrt{4n^4 + 12n^2+9}}{2} \;=\;\frac{(2n^2-1) \pm\sqrt{(2n^2+3)^2}}{2}\)

. . \(\displaystyle k \;=\;\frac{-(2n^2-1) \pm(2n^2+3)}{2} \quad\Rightarrow\quad k \:=\: 2,\;\;2n^2-1\) .**


\(\displaystyle \text{Therefore: }\:\boxed{k \:=\:2}\)


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


** . Had an amusing thought . . .

Now that I see the two roots, I can write the factors of the quadratic.

I should have said:
. . "Obviously, this factors: .\(\displaystyle \bigg[k - 2\bigg]\,\bigg[k - (2n^2-1)\bigg] \:=\:0\)

. . and ride off into the sunset while the townfolk mutter "Who was that masked man?"
. . . . . . . and the music swells . . . fade to black.