Derive the following identity $1^2+2^2+ ldots + n^2 = fracn(n+1)(2n+1)6$.

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Count the elements of the following set
$$A=(x,y,z): 1leq x,y,z leq n+1, z>maxx,y.
$$
From this derive the following identity:
$$1^2+2^2+ ldots + n^2 = fracn(n+1)(2n+1)6.$$
In the same manner find the formula for $1^k + 2^k + ldots + n^k$ for $k=3,4$.




It is easy to see that $|A| = 1^2 + 2^2 + ldots + n^2$, since from the sum rule we have $$|A| = sum_i=0^n+1 |(x,y,i): 1leq x,y,z leq n+1, i> maxx,y| = sum_i=0^n+1 i^2$$ (as we can choose $x$ and $y$ in $i times i$ ways for each $i$).



However I can't see why is $|A|$ equals $dfracn(n+1)(2n+1)6$.







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  • Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
    – Henning Makholm
    Aug 20 at 11:27







  • 3




    Your set $A=(x,y,z): 1geq x,y,z geq n+1, z>maxx,y$ does not make sense. Please rewrite it.
    – MathOverview
    Aug 20 at 11:27















up vote
1
down vote

favorite













Count the elements of the following set
$$A=(x,y,z): 1leq x,y,z leq n+1, z>maxx,y.
$$
From this derive the following identity:
$$1^2+2^2+ ldots + n^2 = fracn(n+1)(2n+1)6.$$
In the same manner find the formula for $1^k + 2^k + ldots + n^k$ for $k=3,4$.




It is easy to see that $|A| = 1^2 + 2^2 + ldots + n^2$, since from the sum rule we have $$|A| = sum_i=0^n+1 |(x,y,i): 1leq x,y,z leq n+1, i> maxx,y| = sum_i=0^n+1 i^2$$ (as we can choose $x$ and $y$ in $i times i$ ways for each $i$).



However I can't see why is $|A|$ equals $dfracn(n+1)(2n+1)6$.







share|cite|improve this question






















  • Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
    – Henning Makholm
    Aug 20 at 11:27







  • 3




    Your set $A=(x,y,z): 1geq x,y,z geq n+1, z>maxx,y$ does not make sense. Please rewrite it.
    – MathOverview
    Aug 20 at 11:27













up vote
1
down vote

favorite









up vote
1
down vote

favorite












Count the elements of the following set
$$A=(x,y,z): 1leq x,y,z leq n+1, z>maxx,y.
$$
From this derive the following identity:
$$1^2+2^2+ ldots + n^2 = fracn(n+1)(2n+1)6.$$
In the same manner find the formula for $1^k + 2^k + ldots + n^k$ for $k=3,4$.




It is easy to see that $|A| = 1^2 + 2^2 + ldots + n^2$, since from the sum rule we have $$|A| = sum_i=0^n+1 |(x,y,i): 1leq x,y,z leq n+1, i> maxx,y| = sum_i=0^n+1 i^2$$ (as we can choose $x$ and $y$ in $i times i$ ways for each $i$).



However I can't see why is $|A|$ equals $dfracn(n+1)(2n+1)6$.







share|cite|improve this question















Count the elements of the following set
$$A=(x,y,z): 1leq x,y,z leq n+1, z>maxx,y.
$$
From this derive the following identity:
$$1^2+2^2+ ldots + n^2 = fracn(n+1)(2n+1)6.$$
In the same manner find the formula for $1^k + 2^k + ldots + n^k$ for $k=3,4$.




It is easy to see that $|A| = 1^2 + 2^2 + ldots + n^2$, since from the sum rule we have $$|A| = sum_i=0^n+1 |(x,y,i): 1leq x,y,z leq n+1, i> maxx,y| = sum_i=0^n+1 i^2$$ (as we can choose $x$ and $y$ in $i times i$ ways for each $i$).



However I can't see why is $|A|$ equals $dfracn(n+1)(2n+1)6$.









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edited Aug 20 at 11:45









Bernard

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asked Aug 20 at 11:17









Jack Blackwell

966




966











  • Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
    – Henning Makholm
    Aug 20 at 11:27







  • 3




    Your set $A=(x,y,z): 1geq x,y,z geq n+1, z>maxx,y$ does not make sense. Please rewrite it.
    – MathOverview
    Aug 20 at 11:27

















  • Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
    – Henning Makholm
    Aug 20 at 11:27







  • 3




    Your set $A=(x,y,z): 1geq x,y,z geq n+1, z>maxx,y$ does not make sense. Please rewrite it.
    – MathOverview
    Aug 20 at 11:27
















Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
– Henning Makholm
Aug 20 at 11:27





Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
– Henning Makholm
Aug 20 at 11:27





3




3




Your set $A=(x,y,z): 1geq x,y,z geq n+1, z>maxx,y$ does not make sense. Please rewrite it.
– MathOverview
Aug 20 at 11:27





Your set $A=(x,y,z): 1geq x,y,z geq n+1, z>maxx,y$ does not make sense. Please rewrite it.
– MathOverview
Aug 20 at 11:27











3 Answers
3






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up vote
2
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accepted










On the one hand, $|A|=sum_i=1^n i^2$.

On the other hand, we can also count the elements of $A$ by grouping them according to the order of $x$ and $y$.



  • when $xneq y$ means that $x<y<z$ or $y<x<z$, so it contributes $2binomn+13$.

  • when $x=y$, the amount of $(x,x,z)$ is $binomn+12$.

So $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$






share|cite|improve this answer






















  • This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
    – Henning Makholm
    Aug 20 at 11:30











  • For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
    – Henning Makholm
    Aug 20 at 11:34











  • Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
    – Jack Blackwell
    Aug 20 at 11:42










  • @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
    – rsy56640
    Aug 20 at 11:49











  • Thanks, now it is clear.
    – Jack Blackwell
    Aug 20 at 11:51

















up vote
0
down vote













We can prove it by induction:



Step 1: check it for base case (n=1)



If we replace $n$ by one, we get: $$ 1^2=frac1times 2 times 36$$
which holds $checkmark$



Step 2: assume it is true for $n=k$ and check if holds for $n=k+1$



If it holds for $n=k$, then we have:



$$1^2+2^2+…+k^2=frack(k+1)(2k+1)6$$



Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:



$$1^2+2^2+…+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$



or



$$1^2+2^2+…+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$



(I want to skip some parts here, but its straight forward). Now the right hand side can be reshaped as:



$$frac(k+1)(k+2)(2k+3)6=frack(k+1)(2k+1)6+(k+1)^2$$



which proves it holds for $n=k+1$ if it holds for $n=k$ $checkmark$



This completes the proof.






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    One method is this: When n= 0, the sum is 0. When n= 1, the sum is 1. When n= 2, the sum is 1+ 4= 5. When n= 3, the sum is 1+ 4+ 9= 14. When n= 4 the sum is 1+ 4+ 9+ 16= 30. When n= 5 the sum is 1+ 4+ 9+ 16+ 25= 55, etc.



    Obviously, at each step we are just adding the next square so the "first differences" are: n= 0, 1- 0= 1; n= 1, 5- 1= 4; n= 2, 14- 5= 9; n= 3, 30- 14= 16; n= 4, 55- 30= 25, etc.



    The "second differences" are: n= 0, 4- 1= 3; n= 1, 9- 4= 5; n= 2, 16- 9= 7; n= 3, 25- 16= 9.



    The "third differences" are: n= 0, 5- 3= 2; n= 1, 7- 5= 2; n= 2, 9- 7= 2.



    The third differences are the constant 2 so all succeeding differences are 0. By "Newton's divided difference formula" the sequence is given by $a_n= 0+ 1(n)+ 3(n)(n-1)/2!+ 2(n)(n-1)(n-2)/3!= frac16left(6n+ 9n^2- 9n+ 2n^3- 6n^2+ 4nright)= frac16left(2n^3+ 3n^2+ nright)= frac16n(n+1)(2n+1)$.



    As a check, when n= 1, that is $frac16(2+ 3+ 1)= 1$; when n= 2, that is $frac16(16+ 12+ 2)= 5$; when n= 3, that is $frac16(54+ 27+ 3)= 14$, etc.






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      3 Answers
      3






      active

      oldest

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      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      2
      down vote



      accepted










      On the one hand, $|A|=sum_i=1^n i^2$.

      On the other hand, we can also count the elements of $A$ by grouping them according to the order of $x$ and $y$.



      • when $xneq y$ means that $x<y<z$ or $y<x<z$, so it contributes $2binomn+13$.

      • when $x=y$, the amount of $(x,x,z)$ is $binomn+12$.

      So $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$






      share|cite|improve this answer






















      • This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
        – Henning Makholm
        Aug 20 at 11:30











      • For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
        – Henning Makholm
        Aug 20 at 11:34











      • Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
        – Jack Blackwell
        Aug 20 at 11:42










      • @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
        – rsy56640
        Aug 20 at 11:49











      • Thanks, now it is clear.
        – Jack Blackwell
        Aug 20 at 11:51














      up vote
      2
      down vote



      accepted










      On the one hand, $|A|=sum_i=1^n i^2$.

      On the other hand, we can also count the elements of $A$ by grouping them according to the order of $x$ and $y$.



      • when $xneq y$ means that $x<y<z$ or $y<x<z$, so it contributes $2binomn+13$.

      • when $x=y$, the amount of $(x,x,z)$ is $binomn+12$.

      So $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$






      share|cite|improve this answer






















      • This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
        – Henning Makholm
        Aug 20 at 11:30











      • For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
        – Henning Makholm
        Aug 20 at 11:34











      • Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
        – Jack Blackwell
        Aug 20 at 11:42










      • @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
        – rsy56640
        Aug 20 at 11:49











      • Thanks, now it is clear.
        – Jack Blackwell
        Aug 20 at 11:51












      up vote
      2
      down vote



      accepted







      up vote
      2
      down vote



      accepted






      On the one hand, $|A|=sum_i=1^n i^2$.

      On the other hand, we can also count the elements of $A$ by grouping them according to the order of $x$ and $y$.



      • when $xneq y$ means that $x<y<z$ or $y<x<z$, so it contributes $2binomn+13$.

      • when $x=y$, the amount of $(x,x,z)$ is $binomn+12$.

      So $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$






      share|cite|improve this answer














      On the one hand, $|A|=sum_i=1^n i^2$.

      On the other hand, we can also count the elements of $A$ by grouping them according to the order of $x$ and $y$.



      • when $xneq y$ means that $x<y<z$ or $y<x<z$, so it contributes $2binomn+13$.

      • when $x=y$, the amount of $(x,x,z)$ is $binomn+12$.

      So $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Aug 20 at 11:38

























      answered Aug 20 at 11:28









      rsy56640

      865




      865











      • This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
        – Henning Makholm
        Aug 20 at 11:30











      • For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
        – Henning Makholm
        Aug 20 at 11:34











      • Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
        – Jack Blackwell
        Aug 20 at 11:42










      • @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
        – rsy56640
        Aug 20 at 11:49











      • Thanks, now it is clear.
        – Jack Blackwell
        Aug 20 at 11:51
















      • This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
        – Henning Makholm
        Aug 20 at 11:30











      • For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
        – Henning Makholm
        Aug 20 at 11:34











      • Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
        – Jack Blackwell
        Aug 20 at 11:42










      • @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
        – rsy56640
        Aug 20 at 11:49











      • Thanks, now it is clear.
        – Jack Blackwell
        Aug 20 at 11:51















      This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
      – Henning Makholm
      Aug 20 at 11:30





      This answer looks like it does contain parts of a meaningful (and actually rather clever) argument, but it also looks like you have only written down a small number of the steps you have in your head. Could you expand it with more details so it is clearer what's going on?
      – Henning Makholm
      Aug 20 at 11:30













      For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
      – Henning Makholm
      Aug 20 at 11:34





      For example, start with: "We can also count the elements of $A$ by grouping them according to what the set $x,y,zsubseteq1,2,ldots,n$ is..."
      – Henning Makholm
      Aug 20 at 11:34













      Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
      – Jack Blackwell
      Aug 20 at 11:42




      Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
      – Jack Blackwell
      Aug 20 at 11:42












      @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
      – rsy56640
      Aug 20 at 11:49





      @JackBlackwell For example, when $x=y$, we take 2 elements from $1,2,..,n+1$, name it $i,j$ and suppose $i<j$, so $(i,i,j)$ is element of $A$.
      – rsy56640
      Aug 20 at 11:49













      Thanks, now it is clear.
      – Jack Blackwell
      Aug 20 at 11:51




      Thanks, now it is clear.
      – Jack Blackwell
      Aug 20 at 11:51










      up vote
      0
      down vote













      We can prove it by induction:



      Step 1: check it for base case (n=1)



      If we replace $n$ by one, we get: $$ 1^2=frac1times 2 times 36$$
      which holds $checkmark$



      Step 2: assume it is true for $n=k$ and check if holds for $n=k+1$



      If it holds for $n=k$, then we have:



      $$1^2+2^2+…+k^2=frack(k+1)(2k+1)6$$



      Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:



      $$1^2+2^2+…+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$



      or



      $$1^2+2^2+…+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$



      (I want to skip some parts here, but its straight forward). Now the right hand side can be reshaped as:



      $$frac(k+1)(k+2)(2k+3)6=frack(k+1)(2k+1)6+(k+1)^2$$



      which proves it holds for $n=k+1$ if it holds for $n=k$ $checkmark$



      This completes the proof.






      share|cite|improve this answer
























        up vote
        0
        down vote













        We can prove it by induction:



        Step 1: check it for base case (n=1)



        If we replace $n$ by one, we get: $$ 1^2=frac1times 2 times 36$$
        which holds $checkmark$



        Step 2: assume it is true for $n=k$ and check if holds for $n=k+1$



        If it holds for $n=k$, then we have:



        $$1^2+2^2+…+k^2=frack(k+1)(2k+1)6$$



        Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:



        $$1^2+2^2+…+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$



        or



        $$1^2+2^2+…+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$



        (I want to skip some parts here, but its straight forward). Now the right hand side can be reshaped as:



        $$frac(k+1)(k+2)(2k+3)6=frack(k+1)(2k+1)6+(k+1)^2$$



        which proves it holds for $n=k+1$ if it holds for $n=k$ $checkmark$



        This completes the proof.






        share|cite|improve this answer






















          up vote
          0
          down vote










          up vote
          0
          down vote









          We can prove it by induction:



          Step 1: check it for base case (n=1)



          If we replace $n$ by one, we get: $$ 1^2=frac1times 2 times 36$$
          which holds $checkmark$



          Step 2: assume it is true for $n=k$ and check if holds for $n=k+1$



          If it holds for $n=k$, then we have:



          $$1^2+2^2+…+k^2=frack(k+1)(2k+1)6$$



          Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:



          $$1^2+2^2+…+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$



          or



          $$1^2+2^2+…+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$



          (I want to skip some parts here, but its straight forward). Now the right hand side can be reshaped as:



          $$frac(k+1)(k+2)(2k+3)6=frack(k+1)(2k+1)6+(k+1)^2$$



          which proves it holds for $n=k+1$ if it holds for $n=k$ $checkmark$



          This completes the proof.






          share|cite|improve this answer












          We can prove it by induction:



          Step 1: check it for base case (n=1)



          If we replace $n$ by one, we get: $$ 1^2=frac1times 2 times 36$$
          which holds $checkmark$



          Step 2: assume it is true for $n=k$ and check if holds for $n=k+1$



          If it holds for $n=k$, then we have:



          $$1^2+2^2+…+k^2=frack(k+1)(2k+1)6$$



          Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:



          $$1^2+2^2+…+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$



          or



          $$1^2+2^2+…+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$



          (I want to skip some parts here, but its straight forward). Now the right hand side can be reshaped as:



          $$frac(k+1)(k+2)(2k+3)6=frack(k+1)(2k+1)6+(k+1)^2$$



          which proves it holds for $n=k+1$ if it holds for $n=k$ $checkmark$



          This completes the proof.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 20 at 11:43









          Alla Tarighati

          2623




          2623




















              up vote
              0
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              One method is this: When n= 0, the sum is 0. When n= 1, the sum is 1. When n= 2, the sum is 1+ 4= 5. When n= 3, the sum is 1+ 4+ 9= 14. When n= 4 the sum is 1+ 4+ 9+ 16= 30. When n= 5 the sum is 1+ 4+ 9+ 16+ 25= 55, etc.



              Obviously, at each step we are just adding the next square so the "first differences" are: n= 0, 1- 0= 1; n= 1, 5- 1= 4; n= 2, 14- 5= 9; n= 3, 30- 14= 16; n= 4, 55- 30= 25, etc.



              The "second differences" are: n= 0, 4- 1= 3; n= 1, 9- 4= 5; n= 2, 16- 9= 7; n= 3, 25- 16= 9.



              The "third differences" are: n= 0, 5- 3= 2; n= 1, 7- 5= 2; n= 2, 9- 7= 2.



              The third differences are the constant 2 so all succeeding differences are 0. By "Newton's divided difference formula" the sequence is given by $a_n= 0+ 1(n)+ 3(n)(n-1)/2!+ 2(n)(n-1)(n-2)/3!= frac16left(6n+ 9n^2- 9n+ 2n^3- 6n^2+ 4nright)= frac16left(2n^3+ 3n^2+ nright)= frac16n(n+1)(2n+1)$.



              As a check, when n= 1, that is $frac16(2+ 3+ 1)= 1$; when n= 2, that is $frac16(16+ 12+ 2)= 5$; when n= 3, that is $frac16(54+ 27+ 3)= 14$, etc.






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                up vote
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                down vote













                One method is this: When n= 0, the sum is 0. When n= 1, the sum is 1. When n= 2, the sum is 1+ 4= 5. When n= 3, the sum is 1+ 4+ 9= 14. When n= 4 the sum is 1+ 4+ 9+ 16= 30. When n= 5 the sum is 1+ 4+ 9+ 16+ 25= 55, etc.



                Obviously, at each step we are just adding the next square so the "first differences" are: n= 0, 1- 0= 1; n= 1, 5- 1= 4; n= 2, 14- 5= 9; n= 3, 30- 14= 16; n= 4, 55- 30= 25, etc.



                The "second differences" are: n= 0, 4- 1= 3; n= 1, 9- 4= 5; n= 2, 16- 9= 7; n= 3, 25- 16= 9.



                The "third differences" are: n= 0, 5- 3= 2; n= 1, 7- 5= 2; n= 2, 9- 7= 2.



                The third differences are the constant 2 so all succeeding differences are 0. By "Newton's divided difference formula" the sequence is given by $a_n= 0+ 1(n)+ 3(n)(n-1)/2!+ 2(n)(n-1)(n-2)/3!= frac16left(6n+ 9n^2- 9n+ 2n^3- 6n^2+ 4nright)= frac16left(2n^3+ 3n^2+ nright)= frac16n(n+1)(2n+1)$.



                As a check, when n= 1, that is $frac16(2+ 3+ 1)= 1$; when n= 2, that is $frac16(16+ 12+ 2)= 5$; when n= 3, that is $frac16(54+ 27+ 3)= 14$, etc.






                share|cite|improve this answer






















                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  One method is this: When n= 0, the sum is 0. When n= 1, the sum is 1. When n= 2, the sum is 1+ 4= 5. When n= 3, the sum is 1+ 4+ 9= 14. When n= 4 the sum is 1+ 4+ 9+ 16= 30. When n= 5 the sum is 1+ 4+ 9+ 16+ 25= 55, etc.



                  Obviously, at each step we are just adding the next square so the "first differences" are: n= 0, 1- 0= 1; n= 1, 5- 1= 4; n= 2, 14- 5= 9; n= 3, 30- 14= 16; n= 4, 55- 30= 25, etc.



                  The "second differences" are: n= 0, 4- 1= 3; n= 1, 9- 4= 5; n= 2, 16- 9= 7; n= 3, 25- 16= 9.



                  The "third differences" are: n= 0, 5- 3= 2; n= 1, 7- 5= 2; n= 2, 9- 7= 2.



                  The third differences are the constant 2 so all succeeding differences are 0. By "Newton's divided difference formula" the sequence is given by $a_n= 0+ 1(n)+ 3(n)(n-1)/2!+ 2(n)(n-1)(n-2)/3!= frac16left(6n+ 9n^2- 9n+ 2n^3- 6n^2+ 4nright)= frac16left(2n^3+ 3n^2+ nright)= frac16n(n+1)(2n+1)$.



                  As a check, when n= 1, that is $frac16(2+ 3+ 1)= 1$; when n= 2, that is $frac16(16+ 12+ 2)= 5$; when n= 3, that is $frac16(54+ 27+ 3)= 14$, etc.






                  share|cite|improve this answer












                  One method is this: When n= 0, the sum is 0. When n= 1, the sum is 1. When n= 2, the sum is 1+ 4= 5. When n= 3, the sum is 1+ 4+ 9= 14. When n= 4 the sum is 1+ 4+ 9+ 16= 30. When n= 5 the sum is 1+ 4+ 9+ 16+ 25= 55, etc.



                  Obviously, at each step we are just adding the next square so the "first differences" are: n= 0, 1- 0= 1; n= 1, 5- 1= 4; n= 2, 14- 5= 9; n= 3, 30- 14= 16; n= 4, 55- 30= 25, etc.



                  The "second differences" are: n= 0, 4- 1= 3; n= 1, 9- 4= 5; n= 2, 16- 9= 7; n= 3, 25- 16= 9.



                  The "third differences" are: n= 0, 5- 3= 2; n= 1, 7- 5= 2; n= 2, 9- 7= 2.



                  The third differences are the constant 2 so all succeeding differences are 0. By "Newton's divided difference formula" the sequence is given by $a_n= 0+ 1(n)+ 3(n)(n-1)/2!+ 2(n)(n-1)(n-2)/3!= frac16left(6n+ 9n^2- 9n+ 2n^3- 6n^2+ 4nright)= frac16left(2n^3+ 3n^2+ nright)= frac16n(n+1)(2n+1)$.



                  As a check, when n= 1, that is $frac16(2+ 3+ 1)= 1$; when n= 2, that is $frac16(16+ 12+ 2)= 5$; when n= 3, that is $frac16(54+ 27+ 3)= 14$, etc.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Aug 20 at 11:48









                  user247327

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