You know those questions we have all seen on standardized tests. You remember the kind, "Find the next number in the sequence 2, 4, 6, 8, . . . ." This note talks about problems like that.

I don't like these kinds of problems at all because any number can be a valid answer.

Consider the problem I mentioned at the start (2, 4, 6, 8, ...), where the "obvious" correct answer is 10. But suppose I want the answer to be 17.4. Then the following model will give a sequence beginning with 2, 4, 6, 8, and 17.4.

 f(x) = 2(x-2)(x-3)(x-4)(x-5)/24
         + 4(x-1)(x-3)(x-4)(x-5)/(-6)
           + 6(x-1)(x-2)(x-4)(x-5)/4
             + 8(x-1)(x-2)(x-3)(x-5)/(-6)
               + 17.4(x-1)(x-2)(x-3)(x-4)/24

To find the first term, evaluate the function at 1. That is, wherever you see an x put a 1 and simplify. You'll get

 f(1) = 2(1-2)(1-3)(1-4)(1-5)/24
         + 4(1-1)(1-3)(1-4)(1-5)/(-6)
           + 6(1-1)(1-2)(1-4)(1-5)/4
             + 8(1-1)(1-2)(1-3)(1-5)/(-6)
               + 17.4(1-1)(1-2)(1-3)(1-4)/24 
       = 2(24)/24 + 0 + 0 + 0 + 0 = 2

To find the second term, evaluate the function at 2. That is, wherever you see an x put a 2 and simplify. You'll get

 f(2) = 2(2-2)(2-3)(2-4)(2-5)/24
         + 4(2-1)(2-3)(2-4)(2-5)/(-6)
           + 6(2-1)(2-2)(2-4)(2-5)/4
             + 8(2-1)(2-2)(2-3)(2-5)/(-6)
               + 17.4(2-1)(2-2)(2-3)(2-4)/24 
       = 0 + 4(-6)/(-6) + 0 + 0 + 0 = 4

To find the third term, evaluate the function at 3. That is, wherever you see an x put a 3 and simplify. You'll get

 f(3) = 2(3-2)(3-3)(3-4)(3-5)/24
         + 4(3-1)(3-3)(3-4)(3-5)/(-6)
           + 6(3-1)(3-2)(3-4)(3-5)/4
             + 8(3-1)(3-2)(3-3)(3-5)/(-6)
               + 17.4(3-1)(3-2)(3-3)(3-4)/24 
       = 0 + 0 + 6(4)/4  + 0 + 0 = 6

To find the fourth term, evaluate the function at 4. That is, wherever you see an x put a 4 and simplify. You'll get

 f(4) = 2(4-2)(4-3)(4-4)(4-5)/24
         + 4(4-1)(4-3)(4-4)(4-5)/(-6)
           + 6(4-1)(4-2)(4-4)(4-5)/4
             + 8(4-1)(4-2)(4-3)(4-5)/(-6)
               + 17.4(4-1)(4-2)(4-3)(4-4)/24 
       = 0 + 0 + 0 + 8(-6)/(-6) + 0 + 0 = 8

To find the fifth term, evaluate the function at 5. That is, wherever you see an x put a 5 and simplify. You'll get

 f(5) = 2(5-2)(5-3)(5-4)(5-5)/24
         + 4(5-1)(5-3)(5-4)(5-5)/(-6)
           + 6(5-1)(5-2)(5-4)(5-5)/4
             + 8(5-1)(5-2)(5-3)(5-5)/(-6)
               + 17.4(5-1)(5-2)(5-3)(5-4)/24 
       = 0 + 0 + 0 + 0 + 17.4(24)/24 = 17.4

Adding Any Two Numbers to a Sequence

Problem: Find the next two terms of the sequence 14, 17, 50, 25, ...

Choose any two numbers you want. Call them A and B. Define f(x) as follows:

 f(x) = 14(x-2)(x-3)(x-4)(x-5)(x-6)/(-120)
         + 17(x-1)(x-3)(x-4)(x-5)(x-6)/24
           + 50(x-1)(x-2)(x-4)(x-5)(x-6)/(-12)
             + 25(x-1)(x-2)(x-3)(x-5)(x-6)/12
               + A(x-1)(x-2)(x-3)(x-4)(x-6)/(-24)
                 + B(x-1)(x-2)(x-3)(x-4)(x-5)/120

Then f(1) = 14, f(2) = 17, f(3) = 50, f(4) = 25, f(5) = A and f(6) = B. So any answer you want, even complex numbers, will work.