if 8=56, 7=42, 6=30, 5=n20 then what does 3 equal in the progressio

Question

Brain teaser question that was posed to me by a friend

Answer 1

It turns out that 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)

Answer 2

I think this is right except 4==>12. and not 8. 

Answer 3

if n is a typo, then wouldn't the answer be 6?

 

8 ==>  56

'                    )  14

7 ==>  42                )2

'                    )  12

6 ==>  30                )2

'                    ) 10

5 ==>  20                )2

'                    )  8

4 ==>  8                  )2          oops this should have been 4 ==>  12; thank you Pranjali.

'                    )  6

3 ==>  6

 

8 x 7 =56, 7 x 6 = 42, etc.

 

Should we take time to find an equation?  Note the second differences are the same, so it is quadratic:

y = ax^2  + bx  + c

 

56 = a8^2  + b8  + c  ---->  64a  +  8b  + c = 56

42 = a7^2  + b7  + c  ---->  49a  +  7b  + c = 42

30 = a6^2  + b6  + c  ---->  36a  +  6b  + c = 30

 

Solving these simultaneous equations gives a = 1, b = -1 and c = 0.

 

So the quadratic equation is y = 1 x^2  -  x   +  0

 

i.e.   y = x^2  -   x

 

or    y  = x (x  -   1)

 

Chk:  sub. 8, 7, 6, etc. for x:

 

       y = 8 (8 - 1) = 8 times 7 = 56

       y = 7 (7 - 1) = 7 times 6 = 42

       y = 6 (6 - 1) = 6 times 5 = 30

etc.