What is the equation to find the 21th number of the sequence: 4, 6, 10, 16, ...
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What is the equation to find the 21th number of the sequence: 4, 6, 10, 16, ...
The 2nd DIFFERENCES (2) are the same, so we have a QUADRATIC sequence. We then use 
the quadratic form of an equation to find the required equation. 
QUADRATIC equation form: 
We can use any 3 points. 1st point is 4, so coordinate point is (x1, y1) = (1, 4)
                                            2nd point is 6, so coordinate point is (x2, y2) = (2, 6)
                                            3rd point is 10, so coordinate point is (x3, y3) = (3, 10)

                   (1, 4)                                               (2, 6)                                                        (3, 10)
                                                       
    4 = A + B + C ---- eq (i)            6 = 4A + 2B + C ---- eq (ii)                  10 = 9A + 3B + C ---- eq (iii)        

      4 =  A +    B + C --- eq (i)
      6 = 4A + 2B + C --- eq (ii)
    10 = 9A + 3B + C --- eq (iii) 
      2 = 3A + B ------ Subtracting eq (i) from eq (ii) ---- eq (iv)
      4 = 5A + B  ----- Subtracting eq (ii) from eq (iii) --- eq (v)
      2 = 2A ----- Subtracting eq (iv) from eq (v)
      

      2 = 3(1) = B ------ Substituting 1 for A in eq (iv)
2 - 3 = B
   - 1 = B

     4 = 1 + - 1 + C ---- Substituting 1 for A, and - 1 for B, in eq (i)
     4 = C

With A being 1, B being - 1, and C being 4, the equation for this sequence is:
 = 

So, the 21st term in this sequence, or