Compute the following sum.












1














I want to compute the following sum
$$S = sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor.$$
Here is what I tried:
$$ S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor.$$
If $m= 2t$ then
$$S =sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{(t-1)t}{2} = t^2.$$
If $m= 2t+1$ then
$$S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{t(t+1)}{2}= t(t+1).$$



But I am not sure if this is correct. Perhaps someone could give an indication.






sequences-and-series number-theory







share|cite|improve this question


















  • 2




    It's incorrect since $S$ is a function of $m$, not $t$.
    – Kemono Chen
    2 hours ago










  • You can write $t$ in terms of $m$ in each case.
    – Hello_World
    2 hours ago










  • @Hello_World Actuallly, you can.
    – 5xum
    2 hours ago










  • Are you asking me to write them in terms of $m$? I can do that if it helps.
    – Hello_World
    2 hours ago


















1














I want to compute the following sum
$$S = sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor.$$
Here is what I tried:
$$ S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor.$$
If $m= 2t$ then
$$S =sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{(t-1)t}{2} = t^2.$$
If $m= 2t+1$ then
$$S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{t(t+1)}{2}= t(t+1).$$



But I am not sure if this is correct. Perhaps someone could give an indication.






sequences-and-series number-theory







share|cite|improve this question


















  • 2




    It's incorrect since $S$ is a function of $m$, not $t$.
    – Kemono Chen
    2 hours ago










  • You can write $t$ in terms of $m$ in each case.
    – Hello_World
    2 hours ago










  • @Hello_World Actuallly, you can.
    – 5xum
    2 hours ago










  • Are you asking me to write them in terms of $m$? I can do that if it helps.
    – Hello_World
    2 hours ago
















1












1








1







I want to compute the following sum
$$S = sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor.$$
Here is what I tried:
$$ S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor.$$
If $m= 2t$ then
$$S =sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{(t-1)t}{2} = t^2.$$
If $m= 2t+1$ then
$$S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{t(t+1)}{2}= t(t+1).$$



But I am not sure if this is correct. Perhaps someone could give an indication.






sequences-and-series number-theory







share|cite|improve this question













I want to compute the following sum
$$S = sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor.$$
Here is what I tried:
$$ S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor.$$
If $m= 2t$ then
$$S =sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{(t-1)t}{2} = t^2.$$
If $m= 2t+1$ then
$$S = sum_{kgeq 0, 2|k}^{m} leftlfloor frac{k}{2}rightrfloor + sum_{kgeq 0, 2not |k}^{m} leftlfloor frac{k}{2}rightrfloor = frac{t(t+1)}{2} + frac{t(t+1)}{2}= t(t+1).$$



But I am not sure if this is correct. Perhaps someone could give an indication.






sequences-and-series number-theory




sequences-and-series number-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









Hello_WorldHello_World

4,14521630




4,14521630








  • 2




    It's incorrect since $S$ is a function of $m$, not $t$.
    – Kemono Chen
    2 hours ago










  • You can write $t$ in terms of $m$ in each case.
    – Hello_World
    2 hours ago










  • @Hello_World Actuallly, you can.
    – 5xum
    2 hours ago










  • Are you asking me to write them in terms of $m$? I can do that if it helps.
    – Hello_World
    2 hours ago
















  • 2




    It's incorrect since $S$ is a function of $m$, not $t$.
    – Kemono Chen
    2 hours ago










  • You can write $t$ in terms of $m$ in each case.
    – Hello_World
    2 hours ago










  • @Hello_World Actuallly, you can.
    – 5xum
    2 hours ago










  • Are you asking me to write them in terms of $m$? I can do that if it helps.
    – Hello_World
    2 hours ago










2




2




It's incorrect since $S$ is a function of $m$, not $t$.
– Kemono Chen
2 hours ago




It's incorrect since $S$ is a function of $m$, not $t$.
– Kemono Chen
2 hours ago












You can write $t$ in terms of $m$ in each case.
– Hello_World
2 hours ago




You can write $t$ in terms of $m$ in each case.
– Hello_World
2 hours ago












@Hello_World Actuallly, you can.
– 5xum
2 hours ago




@Hello_World Actuallly, you can.
– 5xum
2 hours ago












Are you asking me to write them in terms of $m$? I can do that if it helps.
– Hello_World
2 hours ago






Are you asking me to write them in terms of $m$? I can do that if it helps.
– Hello_World
2 hours ago












1 Answer
1






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oldest

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5














Yes, you are correct. You may also write the result as a more compact formula:
$$sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor=
begin{cases}
t^2&text {if $m=2t$}\
t(t+1)&text {if $m=2t+1$}\
end{cases}=leftlfloor frac{m^2}{4}rightrfloor.$$
Indeed, if $m=2t$ then
$$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2rightrfloor=t^2$$
and if $m=2t+1$ then
$$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2+t+frac{1}{4}rightrfloor=t(t+1).$$






share|cite|improve this answer























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    Yes, you are correct. You may also write the result as a more compact formula:
    $$sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor=
    begin{cases}
    t^2&text {if $m=2t$}\
    t(t+1)&text {if $m=2t+1$}\
    end{cases}=leftlfloor frac{m^2}{4}rightrfloor.$$
    Indeed, if $m=2t$ then
    $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2rightrfloor=t^2$$
    and if $m=2t+1$ then
    $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2+t+frac{1}{4}rightrfloor=t(t+1).$$






    share|cite|improve this answer




























      5














      Yes, you are correct. You may also write the result as a more compact formula:
      $$sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor=
      begin{cases}
      t^2&text {if $m=2t$}\
      t(t+1)&text {if $m=2t+1$}\
      end{cases}=leftlfloor frac{m^2}{4}rightrfloor.$$
      Indeed, if $m=2t$ then
      $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2rightrfloor=t^2$$
      and if $m=2t+1$ then
      $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2+t+frac{1}{4}rightrfloor=t(t+1).$$






      share|cite|improve this answer


























        5












        5








        5






        Yes, you are correct. You may also write the result as a more compact formula:
        $$sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor=
        begin{cases}
        t^2&text {if $m=2t$}\
        t(t+1)&text {if $m=2t+1$}\
        end{cases}=leftlfloor frac{m^2}{4}rightrfloor.$$
        Indeed, if $m=2t$ then
        $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2rightrfloor=t^2$$
        and if $m=2t+1$ then
        $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2+t+frac{1}{4}rightrfloor=t(t+1).$$






        share|cite|improve this answer














        Yes, you are correct. You may also write the result as a more compact formula:
        $$sum_{k=0}^{m} leftlfloor frac{k}{2}rightrfloor=
        begin{cases}
        t^2&text {if $m=2t$}\
        t(t+1)&text {if $m=2t+1$}\
        end{cases}=leftlfloor frac{m^2}{4}rightrfloor.$$
        Indeed, if $m=2t$ then
        $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2rightrfloor=t^2$$
        and if $m=2t+1$ then
        $$leftlfloor frac{m^2}{4}rightrfloor=leftlfloor t^2+t+frac{1}{4}rightrfloor=t(t+1).$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 hours ago

























        answered 2 hours ago









        Robert ZRobert Z

        94.2k1061132




        94.2k1061132






























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