Derive the following identity $1^2+2^2+ ldots + n^2 = fracn(n+1)(2n+1)6$.
Clash Royale CLAN TAG#URR8PPP
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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 $dfracn(n+1)(2n+1)6$.
discrete-mathematics
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up vote
1
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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 $dfracn(n+1)(2n+1)6$.
discrete-mathematics
Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
â 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.
â MathOverview
Aug 20 at 11:27
add a comment |Â
up vote
1
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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 $dfracn(n+1)(2n+1)6$.
discrete-mathematics
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 $dfracn(n+1)(2n+1)6$.
discrete-mathematics
edited Aug 20 at 11:45
Bernard
111k635103
111k635103
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?
â 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.
â MathOverview
Aug 20 at 11:27
add a comment |Â
Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
â 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.
â MathOverview
Aug 20 at 11:27
Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
â 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?
â 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.
â 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.
â MathOverview
Aug 20 at 11:27
add a comment |Â
3 Answers
3
active
oldest
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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 $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$
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?
â 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..."
â Henning Makholm
Aug 20 at 11:34
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â 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 $A$.
â rsy56640
Aug 20 at 11:49
Thanks, now it is clear.
â Jack Blackwell
Aug 20 at 11:51
add a comment |Â
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:
$$1^2+2^2+â¦+k^2=frack(k+1)(2k+1)6$$
Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:
$$1^2+2^2+â¦+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$
or
$$1^2+2^2+â¦+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$
(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.
add a comment |Â
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.
add a comment |Â
3 Answers
3
active
oldest
votes
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 $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$
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?
â 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..."
â Henning Makholm
Aug 20 at 11:34
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â 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 $A$.
â rsy56640
Aug 20 at 11:49
Thanks, now it is clear.
â Jack Blackwell
Aug 20 at 11:51
add a comment |Â
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 $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$
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?
â 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..."
â Henning Makholm
Aug 20 at 11:34
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â 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 $A$.
â rsy56640
Aug 20 at 11:49
Thanks, now it is clear.
â Jack Blackwell
Aug 20 at 11:51
add a comment |Â
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 $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$
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 $sum_i=1^n i^2=|A|=2binomn+13+binomn+12=fracn(n+1)(2n+1)6$
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?
â 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..."
â Henning Makholm
Aug 20 at 11:34
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â 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 $A$.
â rsy56640
Aug 20 at 11:49
Thanks, now it is clear.
â Jack Blackwell
Aug 20 at 11:51
add a comment |Â
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?
â 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..."
â Henning Makholm
Aug 20 at 11:34
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â 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 $A$.
â rsy56640
Aug 20 at 11:49
Thanks, now it is clear.
â 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?
â 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?
â 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..."
â 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..."
â Henning Makholm
Aug 20 at 11:34
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â Jack Blackwell
Aug 20 at 11:42
Can You please write, why the two cases contributes to $n+1choose3$ and $n+1choose2$ respectively?
â 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 $A$.
â 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 $A$.
â rsy56640
Aug 20 at 11:49
Thanks, now it is clear.
â Jack Blackwell
Aug 20 at 11:51
Thanks, now it is clear.
â Jack Blackwell
Aug 20 at 11:51
add a comment |Â
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:
$$1^2+2^2+â¦+k^2=frack(k+1)(2k+1)6$$
Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:
$$1^2+2^2+â¦+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$
or
$$1^2+2^2+â¦+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$
(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.
add a comment |Â
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:
$$1^2+2^2+â¦+k^2=frack(k+1)(2k+1)6$$
Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:
$$1^2+2^2+â¦+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$
or
$$1^2+2^2+â¦+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$
(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.
add a comment |Â
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:
$$1^2+2^2+â¦+k^2=frack(k+1)(2k+1)6$$
Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:
$$1^2+2^2+â¦+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$
or
$$1^2+2^2+â¦+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$
(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.
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:
$$1^2+2^2+â¦+k^2=frack(k+1)(2k+1)6$$
Now we show it also holds for $n=k+1$. By replacing $n$ by $k+1$ we should show:
$$1^2+2^2+â¦+(k+1)^2=frac(k+1)(k+2)(2(k+1)+1)6$$
or
$$1^2+2^2+â¦+k^2+(k+1)^2=frac(k+1)(k+2)(2k+3)6$$
(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.
answered Aug 20 at 11:43
Alla Tarighati
2623
2623
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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.
add a comment |Â
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.
add a comment |Â
up vote
0
down vote
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.
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.
answered Aug 20 at 11:48
user247327
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Shouldn't one of your bounds on $z$ (in the definition of $A$) be the other way around?
â 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.
â MathOverview
Aug 20 at 11:27