SOLUTION: Try to solve this sequence 112 , 2112 , 2122 .... Calculate the further terms

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Question 558951: Try to solve this sequence
112 , 2112 , 2122 ....
Calculate the further terms
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
t1 = 112
t2 = 2112
t3 = 2122

Three terms always determine a quadratic sequence.

So let the nth term be 

tn = an² + bn + c

Then 

t1 = 112 = a(1)² + b(1) + c 

                          a + b + c =  112

t2 = 2112 = a(2)² + b(2) + c 

                         4a + 2b + c = 2112

t3 = 2122 = a(3)² + b(3) + c 

                         9a + 3b + c = 2122

 So we solve the system:

ì a +  b + c =   112
í4a + 2b + c =  2112
î9a + 3b + c =  2122

We get a=-995, b = 4985, and c = -3878

So the three terms given fit the sequence:

tn = -995n² + 4985t - 3878

So the first 50 terms are:

t1 = -995(1)² + 4985(1) - 3878 = 112
t2 = -995(2)² + 4985(2) - 3878 = 2112
t3 = -995(3)² + 4985(3) - 3878 = 2122
t4 = -995(4)² + 4985(4) - 3878 = 142
t5 = -995(5)² + 4985(5) - 3878 = -3828
t6 = -995(6)² + 4985(6) - 3878 = -9788
t7 = -995(7)² + 4985(7) - 3878 = -17738
t8 = -995(8)² + 4985(8) - 3878 = -27678
t9 = -995(9)² + 4985(9) - 3878 = -39608
t10 = -995(10)² + 4985(10) - 3878 = -53528
t11 = -995(11)² + 4985(11) - 3878 = -69438
t12 = -995(12)² + 4985(12) - 3878 = -87338
t13 = -995(13)² + 4985(13) - 3878 = -107228
t14 = -995(14)² + 4985(14) - 3878 = -129108
t15 = -995(15)² + 4985(15) - 3878 = -152978
t16 = -995(16)² + 4985(16) - 3878 = -178838
t17 = -995(17)² + 4985(17) - 3878 = -206688
t18 = -995(18)² + 4985(18) - 3878 = -236528
t19 = -995(19)² + 4985(19) - 3878 = -268358
t20 = -995(20)² + 4985(20) - 3878 = -302178
t21 = -995(21)² + 4985(21) - 3878 = -337988
t22 = -995(22)² + 4985(22) - 3878 = -375788
t23 = -995(23)² + 4985(23) - 3878 = -415578
t24 = -995(24)² + 4985(24) - 3878 = -457358
t25 = -995(25)² + 4985(25) - 3878 = -501128
t26 = -995(26)² + 4985(26) - 3878 = -546888
t27 = -995(27)² + 4985(27) - 3878 = -594638
t28 = -995(28)² + 4985(28) - 3878 = -644378
t29 = -995(29)² + 4985(29) - 3878 = -696108
t30 = -995(30)² + 4985(30) - 3878 = -749828
t31 = -995(31)² + 4985(31) - 3878 = -805538
t32 = -995(32)² + 4985(32) - 3878 = -863238
t33 = -995(33)² + 4985(33) - 3878 = -922928
t34 = -995(34)² + 4985(34) - 3878 = -984608
t35 = -995(35)² + 4985(35) - 3878 = -1048278
t36 = -995(36)² + 4985(36) - 3878 = -1113938
t37 = -995(37)² + 4985(37) - 3878 = -1181588
t38 = -995(38)² + 4985(38) - 3878 = -1251228
t39 = -995(39)² + 4985(39) - 3878 = -1322858
t40 = -995(40)² + 4985(40) - 3878 = -1396478
t41 = -995(41)² + 4985(41) - 3878 = -1472088
t42 = -995(42)² + 4985(42) - 3878 = -1549688
t43 = -995(43)² + 4985(43) - 3878 = -1629278
t44 = -995(44)² + 4985(44) - 3878 = -1710858
t45 = -995(45)² + 4985(45) - 3878 = -1794428
t46 = -995(46)² + 4985(46) - 3878 = -1879988
t47 = -995(47)² + 4985(47) - 3878 = -1967538
t48 = -995(48)² + 4985(48) - 3878 = -2057078
t49 = -995(49)² + 4985(49) - 3878 = -2148608
t50 = -995(50)² + 4985(50) - 3878 = -2242128

Edwin
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