Mixing
Sum with Binomial Coefficients to Floor Function
$begingroup$
all!
I have recently begun my studies in an upper-level discrete mathematics course. So far, I have quite enjoyed all of the lectures.
I recently came across an exercise on a supplemental homework requesting me to compute a trio of three binomial sums containing floor functions:
$$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+1}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+2}$$
About an hour ago, I had not even heard of ceiling and floor functions.
I have tried to work the first sum out, but I really am not sure how to continue with these floor functions, nor am I sure how to truly approach these sums.
Any and all help would be greatly appreciated! Thank you all for taking the time to read my post.
combinatorics discrete-mathematics
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
$endgroup$
add a comment |
$begingroup$
all!
I have recently begun my studies in an upper-level discrete mathematics course. So far, I have quite enjoyed all of the lectures.
I recently came across an exercise on a supplemental homework requesting me to compute a trio of three binomial sums containing floor functions:
$$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+1}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+2}$$
About an hour ago, I had not even heard of ceiling and floor functions.
I have tried to work the first sum out, but I really am not sure how to continue with these floor functions, nor am I sure how to truly approach these sums.
Any and all help would be greatly appreciated! Thank you all for taking the time to read my post.
combinatorics discrete-mathematics
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
$endgroup$
$begingroup$
The floors are not actually a problem here. What do you know about binomials ? do you know z-Transform ?
$endgroup$
– G Cab
Jan 22 at 23:52
add a comment |
$begingroup$
all!
I have recently begun my studies in an upper-level discrete mathematics course. So far, I have quite enjoyed all of the lectures.
I recently came across an exercise on a supplemental homework requesting me to compute a trio of three binomial sums containing floor functions:
$$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+1}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+2}$$
About an hour ago, I had not even heard of ceiling and floor functions.
I have tried to work the first sum out, but I really am not sure how to continue with these floor functions, nor am I sure how to truly approach these sums.
Any and all help would be greatly appreciated! Thank you all for taking the time to read my post.
combinatorics discrete-mathematics
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
$endgroup$
all!
I have recently begun my studies in an upper-level discrete mathematics course. So far, I have quite enjoyed all of the lectures.
I recently came across an exercise on a supplemental homework requesting me to compute a trio of three binomial sums containing floor functions:
$$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+1}$$ $$sum_{k=0}^{leftlfloorfrac{n}{3}rightrfloor} {n choose 3k+2}$$
About an hour ago, I had not even heard of ceiling and floor functions.
I have tried to work the first sum out, but I really am not sure how to continue with these floor functions, nor am I sure how to truly approach these sums.
Any and all help would be greatly appreciated! Thank you all for taking the time to read my post.
combinatorics discrete-mathematics
combinatorics discrete-mathematics
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
asked Jan 22 at 22:13
Daniel TortiDaniel Torti
162
162
$begingroup$
The floors are not actually a problem here. What do you know about binomials ? do you know z-Transform ?
$endgroup$
– G Cab
Jan 22 at 23:52
add a comment |
$begingroup$
The floors are not actually a problem here. What do you know about binomials ? do you know z-Transform ?
$endgroup$
– G Cab
Jan 22 at 23:52
$begingroup$
The floors are not actually a problem here. What do you know about binomials ? do you know z-Transform ?
$endgroup$
– G Cab
Jan 22 at 23:52
$begingroup$
The floors are not actually a problem here. What do you know about binomials ? do you know z-Transform ?
$endgroup$
– G Cab
Jan 22 at 23:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: Let $a_n$ be your first summation, and $b_n,c_n$ be the second and third. Start by computing these exactly for small values of $n$. You should start to notice a general pattern emerging. Describe the pattern exactly, then prove that it holds in general using induction on $n$. Use the base cases
$$
a_0 = 1, b_0=0,c_0=0
$$
and the rules
$$
a_{n+1}=a_n+c_n,qquad b_{n+1}=b_n+a_n,qquad c_{n+1}=c_n+b_n
$$
The rule $a_{n+1}=a_n+c_n$ can be proven by applying Pascal's identity to each summand in $a_{n+1}$, and then splitting into two summations, which will be exactly $a_n$ and $c_n$.
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Let $a_n$ be your first summation, and $b_n,c_n$ be the second and third. Start by computing these exactly for small values of $n$. You should start to notice a general pattern emerging. Describe the pattern exactly, then prove that it holds in general using induction on $n$. Use the base cases
$$
a_0 = 1, b_0=0,c_0=0
$$
and the rules
$$
a_{n+1}=a_n+c_n,qquad b_{n+1}=b_n+a_n,qquad c_{n+1}=c_n+b_n
$$
The rule $a_{n+1}=a_n+c_n$ can be proven by applying Pascal's identity to each summand in $a_{n+1}$, and then splitting into two summations, which will be exactly $a_n$ and $c_n$.
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
$endgroup$
add a comment |
$begingroup$
Hint: Let $a_n$ be your first summation, and $b_n,c_n$ be the second and third. Start by computing these exactly for small values of $n$. You should start to notice a general pattern emerging. Describe the pattern exactly, then prove that it holds in general using induction on $n$. Use the base cases
$$
a_0 = 1, b_0=0,c_0=0
$$
and the rules
$$
a_{n+1}=a_n+c_n,qquad b_{n+1}=b_n+a_n,qquad c_{n+1}=c_n+b_n
$$
The rule $a_{n+1}=a_n+c_n$ can be proven by applying Pascal's identity to each summand in $a_{n+1}$, and then splitting into two summations, which will be exactly $a_n$ and $c_n$.
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
$endgroup$
add a comment |
$begingroup$
Hint: Let $a_n$ be your first summation, and $b_n,c_n$ be the second and third. Start by computing these exactly for small values of $n$. You should start to notice a general pattern emerging. Describe the pattern exactly, then prove that it holds in general using induction on $n$. Use the base cases
$$
a_0 = 1, b_0=0,c_0=0
$$
and the rules
$$
a_{n+1}=a_n+c_n,qquad b_{n+1}=b_n+a_n,qquad c_{n+1}=c_n+b_n
$$
The rule $a_{n+1}=a_n+c_n$ can be proven by applying Pascal's identity to each summand in $a_{n+1}$, and then splitting into two summations, which will be exactly $a_n$ and $c_n$.
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
$endgroup$
Hint: Let $a_n$ be your first summation, and $b_n,c_n$ be the second and third. Start by computing these exactly for small values of $n$. You should start to notice a general pattern emerging. Describe the pattern exactly, then prove that it holds in general using induction on $n$. Use the base cases
$$
a_0 = 1, b_0=0,c_0=0
$$
and the rules
$$
a_{n+1}=a_n+c_n,qquad b_{n+1}=b_n+a_n,qquad c_{n+1}=c_n+b_n
$$
The rule $a_{n+1}=a_n+c_n$ can be proven by applying Pascal's identity to each summand in $a_{n+1}$, and then splitting into two summations, which will be exactly $a_n$ and $c_n$.
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
answered Jan 22 at 23:53
Mike EarnestMike Earnest
24.3k22151
24.3k22151
add a comment |
add a comment |
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$begingroup$
The floors are not actually a problem here. What do you know about binomials ? do you know z-Transform ?
$endgroup$
– G Cab
Jan 22 at 23:52