Consider the sequence defined by t_n = an² + bn + c. If t₂ = 3, t₄ = 13 and t₇ = 113, show that 3t_n = 17n² – 87n + 115

Here,                                                   ...(i)
Now,                            
          4a + 2b + c = 3                                               ...(ii)
                                                
           16a + 4b + c = 13                                          ...(iii)
                                             
           49a + 7b + c = 113                                         ...(iv)
Subtracting (iii) from (iv), we get   33a + 3b = 100                ...(v)
Subtracting (ii) from (iii), we get    12a + 2b = 10
              6a + b = 5                                                  ...(vi)
Multiplying both sides of (vi) by 3, we get
                 18a + 3b = 15                                               ...(vii)
Subtracting (vii) from (v), we get
                
Using this value of a in (vi), we get
                  
Using values of a and b in (ii), we get

Using values of a, b, c in (i), we get