Angel A. answered • 03/23/16
Tutor
New to Wyzant
e ^ (i*pi) + 1 = 0
The ‘?’ can be whatever we want it to be!
Allow me to illustrate. Take for example, f(x)=(1/40)x^4-(13/20)x^3+(291/40)x^2-(553/20)x+42, and evaluate f(8), f(7), f(6), f(5), f(3). It turns out that:
f(8)=56,
f(7)=42,
f(6)=30,
f(5)=20,
f(3)=9.
Wait, what sorcery is this? Turns out that although the popular rule f(x)=x(x-1), which gives f(x)=6, satisfies the known values in the sequence, that f(x)=(1/40)x^4-(13/20)x^3+(291/40)x^2-(553/20)x+42 also satisfies them–except with a different value of f(3)!
Here’s another one that also works but gives f(3)=12:
f(x)=(1/20)x^4-(13/10)x^3+(271/20)x^2-(543/10)x+84
And here is one where f(3)=π
f(x)=(1/120)(π-6)x^4-(13/60)(π-6)x^3+(1/120)(251π-1386)x^2+(1/60)(3138-533π)x+14(π-6)
Finally, in general if you want the ‘?’=k, i.e., f(3)=k where k is the value of your choice, then
(1/120)(k-6)x^4-(13/60)(k-6)x^3+(1/120)(251k-1386)x^2+(1/60)(3138-533k)x+14(k-6)
For a non-polynomial rule see here: http://i.imgur.com/BHkg0Ad.png
Allow me to illustrate. Take for example, f(x)=(1/40)x^4-(13/20)x^3+(291/40)x^2-(553/20)x+42, and evaluate f(8), f(7), f(6), f(5), f(3). It turns out that:
f(8)=56,
f(7)=42,
f(6)=30,
f(5)=20,
f(3)=9.
Wait, what sorcery is this? Turns out that although the popular rule f(x)=x(x-1), which gives f(x)=6, satisfies the known values in the sequence, that f(x)=(1/40)x^4-(13/20)x^3+(291/40)x^2-(553/20)x+42 also satisfies them–except with a different value of f(3)!
Here’s another one that also works but gives f(3)=12:
f(x)=(1/20)x^4-(13/10)x^3+(271/20)x^2-(543/10)x+84
And here is one where f(3)=π
f(x)=(1/120)(π-6)x^4-(13/60)(π-6)x^3+(1/120)(251π-1386)x^2+(1/60)(3138-533π)x+14(π-6)
Finally, in general if you want the ‘?’=k, i.e., f(3)=k where k is the value of your choice, then
(1/120)(k-6)x^4-(13/60)(k-6)x^3+(1/120)(251k-1386)x^2+(1/60)(3138-533k)x+14(k-6)
For a non-polynomial rule see here: http://i.imgur.com/BHkg0Ad.png
Larry W.
To those who say the answer is 6, why can't it be 9? Who says that we have to account for the missing 4? So the pattern could be the multiply by number (in parentheses), in decreasing order, like this: 8x(7) = 56 7x(6) = 42 6x(5) = 30 5x(4) = 20 The problem then asks what 3 would be, saying nothing about 4, so the next line would read: 3x(3) = 9 This followed the decreasing numbers in parentheses, 7, 6, 5, 4, then 3, giving us the answer of 9.04/26/20