What will be the next term in this mathematical sequence?












3












$begingroup$


What will be next in this series?
$$0, 6, 24, 60, 120, 210, ...$$
I've tried it and noticed that the numbers are multiples of six. But I couldn't make a relation between them.






mathematics pattern calculation-puzzle







share|improve this question











$endgroup$












  • $begingroup$
    the OEIS has three potential answers to this question.
    $endgroup$
    – Don Thousand
    Dec 25 '18 at 16:32
















3












$begingroup$


What will be next in this series?
$$0, 6, 24, 60, 120, 210, ...$$
I've tried it and noticed that the numbers are multiples of six. But I couldn't make a relation between them.






mathematics pattern calculation-puzzle







share|improve this question











$endgroup$












  • $begingroup$
    the OEIS has three potential answers to this question.
    $endgroup$
    – Don Thousand
    Dec 25 '18 at 16:32














3












3








3





$begingroup$


What will be next in this series?
$$0, 6, 24, 60, 120, 210, ...$$
I've tried it and noticed that the numbers are multiples of six. But I couldn't make a relation between them.






mathematics pattern calculation-puzzle







share|improve this question











$endgroup$




What will be next in this series?
$$0, 6, 24, 60, 120, 210, ...$$
I've tried it and noticed that the numbers are multiples of six. But I couldn't make a relation between them.






mathematics pattern calculation-puzzle




mathematics pattern calculation-puzzle






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited Dec 25 '18 at 9:46









Rand al'Thor

71.1k14235471




71.1k14235471










asked Dec 25 '18 at 9:36









Gurbir SinghGurbir Singh

1193




1193












  • $begingroup$
    the OEIS has three potential answers to this question.
    $endgroup$
    – Don Thousand
    Dec 25 '18 at 16:32


















  • $begingroup$
    the OEIS has three potential answers to this question.
    $endgroup$
    – Don Thousand
    Dec 25 '18 at 16:32
















$begingroup$
the OEIS has three potential answers to this question.
$endgroup$
– Don Thousand
Dec 25 '18 at 16:32




$begingroup$
the OEIS has three potential answers to this question.
$endgroup$
– Don Thousand
Dec 25 '18 at 16:32










4 Answers
4






active

oldest

votes


















1












$begingroup$

It seems like




sum of digit series

f(n)=n*sumOfDigits(n-1)*sumOfDigits(n+1)




example:-




f(1)=1*sumOfDigits(1-1)*sumOfDigits(1+1) = 1*0*2 = 0

f(2)=2*sumOfDigits(2-1)*sumOfDigits(2+1) = 2*1*3 = 6

f(3)=3*sumOfDigits(3-1)*sumOfDigits(3+1) = 3*2*4 = 24

f(4)=4*sumOfDigits(4-1)*sumOfDigits(4+1) = 4*3*5 = 60

f(5)=5*sumOfDigits(5-1)*sumOfDigits(5+1) = 5*4*6 = 120

f(6)=6*sumOfDigits(6-1)*sumOfDigits(6+1) = 6*5*7 = 210




Answer:-




f(7)=7*sumOfDigits(7-1)*sumOfDigits(7+1) = 7*6*8 = 336







share|improve this answer









$endgroup$









  • 2




    $begingroup$
    That's copying my solution
    $endgroup$
    – TheSimpliFire
    Dec 28 '18 at 13:27



















8












$begingroup$

It seems like a




cube series.




Namely,




cube of 1 is 1 then 1 - 1 = 0

cube of 2 is 8 then 8 - 2 = 6

cube of 3 is 27 then 27 - 3 = 24

cube of 4 is 64 then 64 - 4 = 60

cube of 5 is 125 then 125 - 5 = 120

cube of 6 is 216 then 216 - 6 = 210

cube of 7 is 343 then 343 - 7 = 336







share|improve this answer











$endgroup$





















    4












    $begingroup$



    • Take differences between terms:




      $6, 18, 36, 60, 90, ...$





    • Notice that these are




      $6$ times the triangular numbers $1, 3, 6, 12, 15, ...$





    So the next difference should be




    $6times 21 = 126$




    and the next term should be




    $210+126=336$.







    share|improve this answer











    $endgroup$









    • 2




      $begingroup$
      The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
      $endgroup$
      – athin
      Dec 25 '18 at 11:07



















    4












    $begingroup$

    Simple Answer:




    The $n$th term is $n(n-1)(n+1)$ and thus the required one is the seventh term giving an answer of $$7(6)(8)=336.$$







    share|improve this answer









    $endgroup$














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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1












      $begingroup$

      It seems like




      sum of digit series

      f(n)=n*sumOfDigits(n-1)*sumOfDigits(n+1)




      example:-




      f(1)=1*sumOfDigits(1-1)*sumOfDigits(1+1) = 1*0*2 = 0

      f(2)=2*sumOfDigits(2-1)*sumOfDigits(2+1) = 2*1*3 = 6

      f(3)=3*sumOfDigits(3-1)*sumOfDigits(3+1) = 3*2*4 = 24

      f(4)=4*sumOfDigits(4-1)*sumOfDigits(4+1) = 4*3*5 = 60

      f(5)=5*sumOfDigits(5-1)*sumOfDigits(5+1) = 5*4*6 = 120

      f(6)=6*sumOfDigits(6-1)*sumOfDigits(6+1) = 6*5*7 = 210




      Answer:-




      f(7)=7*sumOfDigits(7-1)*sumOfDigits(7+1) = 7*6*8 = 336







      share|improve this answer









      $endgroup$









      • 2




        $begingroup$
        That's copying my solution
        $endgroup$
        – TheSimpliFire
        Dec 28 '18 at 13:27
















      1












      $begingroup$

      It seems like




      sum of digit series

      f(n)=n*sumOfDigits(n-1)*sumOfDigits(n+1)




      example:-




      f(1)=1*sumOfDigits(1-1)*sumOfDigits(1+1) = 1*0*2 = 0

      f(2)=2*sumOfDigits(2-1)*sumOfDigits(2+1) = 2*1*3 = 6

      f(3)=3*sumOfDigits(3-1)*sumOfDigits(3+1) = 3*2*4 = 24

      f(4)=4*sumOfDigits(4-1)*sumOfDigits(4+1) = 4*3*5 = 60

      f(5)=5*sumOfDigits(5-1)*sumOfDigits(5+1) = 5*4*6 = 120

      f(6)=6*sumOfDigits(6-1)*sumOfDigits(6+1) = 6*5*7 = 210




      Answer:-




      f(7)=7*sumOfDigits(7-1)*sumOfDigits(7+1) = 7*6*8 = 336







      share|improve this answer









      $endgroup$









      • 2




        $begingroup$
        That's copying my solution
        $endgroup$
        – TheSimpliFire
        Dec 28 '18 at 13:27














      1












      1








      1





      $begingroup$

      It seems like




      sum of digit series

      f(n)=n*sumOfDigits(n-1)*sumOfDigits(n+1)




      example:-




      f(1)=1*sumOfDigits(1-1)*sumOfDigits(1+1) = 1*0*2 = 0

      f(2)=2*sumOfDigits(2-1)*sumOfDigits(2+1) = 2*1*3 = 6

      f(3)=3*sumOfDigits(3-1)*sumOfDigits(3+1) = 3*2*4 = 24

      f(4)=4*sumOfDigits(4-1)*sumOfDigits(4+1) = 4*3*5 = 60

      f(5)=5*sumOfDigits(5-1)*sumOfDigits(5+1) = 5*4*6 = 120

      f(6)=6*sumOfDigits(6-1)*sumOfDigits(6+1) = 6*5*7 = 210




      Answer:-




      f(7)=7*sumOfDigits(7-1)*sumOfDigits(7+1) = 7*6*8 = 336







      share|improve this answer









      $endgroup$



      It seems like




      sum of digit series

      f(n)=n*sumOfDigits(n-1)*sumOfDigits(n+1)




      example:-




      f(1)=1*sumOfDigits(1-1)*sumOfDigits(1+1) = 1*0*2 = 0

      f(2)=2*sumOfDigits(2-1)*sumOfDigits(2+1) = 2*1*3 = 6

      f(3)=3*sumOfDigits(3-1)*sumOfDigits(3+1) = 3*2*4 = 24

      f(4)=4*sumOfDigits(4-1)*sumOfDigits(4+1) = 4*3*5 = 60

      f(5)=5*sumOfDigits(5-1)*sumOfDigits(5+1) = 5*4*6 = 120

      f(6)=6*sumOfDigits(6-1)*sumOfDigits(6+1) = 6*5*7 = 210




      Answer:-




      f(7)=7*sumOfDigits(7-1)*sumOfDigits(7+1) = 7*6*8 = 336








      share|improve this answer












      share|improve this answer



      share|improve this answer










      answered Dec 27 '18 at 13:03









      Vivek KundariyaVivek Kundariya

      263




      263








      • 2




        $begingroup$
        That's copying my solution
        $endgroup$
        – TheSimpliFire
        Dec 28 '18 at 13:27














      • 2




        $begingroup$
        That's copying my solution
        $endgroup$
        – TheSimpliFire
        Dec 28 '18 at 13:27








      2




      2




      $begingroup$
      That's copying my solution
      $endgroup$
      – TheSimpliFire
      Dec 28 '18 at 13:27




      $begingroup$
      That's copying my solution
      $endgroup$
      – TheSimpliFire
      Dec 28 '18 at 13:27











      8












      $begingroup$

      It seems like a




      cube series.




      Namely,




      cube of 1 is 1 then 1 - 1 = 0

      cube of 2 is 8 then 8 - 2 = 6

      cube of 3 is 27 then 27 - 3 = 24

      cube of 4 is 64 then 64 - 4 = 60

      cube of 5 is 125 then 125 - 5 = 120

      cube of 6 is 216 then 216 - 6 = 210

      cube of 7 is 343 then 343 - 7 = 336







      share|improve this answer











      $endgroup$


















        8












        $begingroup$

        It seems like a




        cube series.




        Namely,




        cube of 1 is 1 then 1 - 1 = 0

        cube of 2 is 8 then 8 - 2 = 6

        cube of 3 is 27 then 27 - 3 = 24

        cube of 4 is 64 then 64 - 4 = 60

        cube of 5 is 125 then 125 - 5 = 120

        cube of 6 is 216 then 216 - 6 = 210

        cube of 7 is 343 then 343 - 7 = 336







        share|improve this answer











        $endgroup$
















          8












          8








          8





          $begingroup$

          It seems like a




          cube series.




          Namely,




          cube of 1 is 1 then 1 - 1 = 0

          cube of 2 is 8 then 8 - 2 = 6

          cube of 3 is 27 then 27 - 3 = 24

          cube of 4 is 64 then 64 - 4 = 60

          cube of 5 is 125 then 125 - 5 = 120

          cube of 6 is 216 then 216 - 6 = 210

          cube of 7 is 343 then 343 - 7 = 336







          share|improve this answer











          $endgroup$



          It seems like a




          cube series.




          Namely,




          cube of 1 is 1 then 1 - 1 = 0

          cube of 2 is 8 then 8 - 2 = 6

          cube of 3 is 27 then 27 - 3 = 24

          cube of 4 is 64 then 64 - 4 = 60

          cube of 5 is 125 then 125 - 5 = 120

          cube of 6 is 216 then 216 - 6 = 210

          cube of 7 is 343 then 343 - 7 = 336








          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited Dec 25 '18 at 13:40









          Omega Krypton

          5,4442847




          5,4442847










          answered Dec 25 '18 at 11:11









          Shailendra SharmaShailendra Sharma

          1803




          1803























              4












              $begingroup$



              • Take differences between terms:




                $6, 18, 36, 60, 90, ...$





              • Notice that these are




                $6$ times the triangular numbers $1, 3, 6, 12, 15, ...$





              So the next difference should be




              $6times 21 = 126$




              and the next term should be




              $210+126=336$.







              share|improve this answer











              $endgroup$









              • 2




                $begingroup$
                The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
                $endgroup$
                – athin
                Dec 25 '18 at 11:07
















              4












              $begingroup$



              • Take differences between terms:




                $6, 18, 36, 60, 90, ...$





              • Notice that these are




                $6$ times the triangular numbers $1, 3, 6, 12, 15, ...$





              So the next difference should be




              $6times 21 = 126$




              and the next term should be




              $210+126=336$.







              share|improve this answer











              $endgroup$









              • 2




                $begingroup$
                The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
                $endgroup$
                – athin
                Dec 25 '18 at 11:07














              4












              4








              4





              $begingroup$



              • Take differences between terms:




                $6, 18, 36, 60, 90, ...$





              • Notice that these are




                $6$ times the triangular numbers $1, 3, 6, 12, 15, ...$





              So the next difference should be




              $6times 21 = 126$




              and the next term should be




              $210+126=336$.







              share|improve this answer











              $endgroup$





              • Take differences between terms:




                $6, 18, 36, 60, 90, ...$





              • Notice that these are




                $6$ times the triangular numbers $1, 3, 6, 12, 15, ...$





              So the next difference should be




              $6times 21 = 126$




              and the next term should be




              $210+126=336$.








              share|improve this answer














              share|improve this answer



              share|improve this answer








              edited Dec 25 '18 at 12:10









              Omega Krypton

              5,4442847




              5,4442847










              answered Dec 25 '18 at 9:46









              Rand al'ThorRand al'Thor

              71.1k14235471




              71.1k14235471








              • 2




                $begingroup$
                The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
                $endgroup$
                – athin
                Dec 25 '18 at 11:07














              • 2




                $begingroup$
                The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
                $endgroup$
                – athin
                Dec 25 '18 at 11:07








              2




              2




              $begingroup$
              The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
              $endgroup$
              – athin
              Dec 25 '18 at 11:07




              $begingroup$
              The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531
              $endgroup$
              – athin
              Dec 25 '18 at 11:07











              4












              $begingroup$

              Simple Answer:




              The $n$th term is $n(n-1)(n+1)$ and thus the required one is the seventh term giving an answer of $$7(6)(8)=336.$$







              share|improve this answer









              $endgroup$


















                4












                $begingroup$

                Simple Answer:




                The $n$th term is $n(n-1)(n+1)$ and thus the required one is the seventh term giving an answer of $$7(6)(8)=336.$$







                share|improve this answer









                $endgroup$
















                  4












                  4








                  4





                  $begingroup$

                  Simple Answer:




                  The $n$th term is $n(n-1)(n+1)$ and thus the required one is the seventh term giving an answer of $$7(6)(8)=336.$$







                  share|improve this answer









                  $endgroup$



                  Simple Answer:




                  The $n$th term is $n(n-1)(n+1)$ and thus the required one is the seventh term giving an answer of $$7(6)(8)=336.$$








                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered Dec 25 '18 at 13:45









                  TheSimpliFireTheSimpliFire

                  2,155532




                  2,155532






























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