Question

In: Statistics and Probability

Write a combinatorial proof for 1 n + 2 ( n − 1 ) + 3...

Write a combinatorial proof for 1 n + 2 ( n − 1 ) + 3 ( n − 2 ) + ⋯ + ( n − 1 ) 2 + n 1 = ( n + 2 choose 3 ) .

Solutions

Expert Solution


Related Solutions

1- Show that (n^3+3n^2+3n+1) / (n+1) is O (n2 ). Use the definition and proof of...
1- Show that (n^3+3n^2+3n+1) / (n+1) is O (n2 ). Use the definition and proof of big-O notation. 2- Prove using the definition of Omega notation that either 8 n is Ω (5 n ) or not. please help be with both
3. To begin a proof by contradiction for “If n is even then n+1 is odd,”...
3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true? 4. Prove that the following is not true by finding a counterexample. “The sum of any 3 consecutive integers is even" 5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd 6.  Show an informal Direct Proof for “The sum of 2 even integers is even.” Recursive Definitions 7.  The Fibonacci Sequence is...
3. The Hofstadter Conway sequence is defined by a(1)=a(2)=1 and (for n>2 by) a(n)=a(a(n-1))+a(n-a(n-1)). Write a...
3. The Hofstadter Conway sequence is defined by a(1)=a(2)=1 and (for n>2 by) a(n)=a(a(n-1))+a(n-a(n-1)). Write a function to quickly compute this sequence. >>> [hc(i) for i in range(1,20)] [1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11]
Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>
Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>
proof a circle is divided into n congruent arcs (n ?? 3), the tangents drawn at...
proof a circle is divided into n congruent arcs (n ?? 3), the tangents drawn at the endpoints of these arcs form a regular polygon.
Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...
Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)
Data Structure: 1. Write a program for f(n) = 1^2+2^3+…+n^2. (i^2 = i*i) 2. If you...
Data Structure: 1. Write a program for f(n) = 1^2+2^3+…+n^2. (i^2 = i*i) 2. If you have the following polynomial function f(n)=a0 +a1 x + a2x2+…+an xn , then you are asked to write a program for that, how do you do? 3. Write a function in C++ to sort array A[]. (You can assume that you have 10 elements in the array.) 4. Analyze the following program, tell us what does it do for each location of “???” (...
Which codes add 1 to integer n? 1) n=n+1; 2) n++; 3)++n; 4) n+=1; 5) n=--n+2
Which codes add 1 to integer n? 1) n=n+1; 2) n++; 3)++n; 4) n+=1; 5) n=--n+2
(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an epsilon, N argument to show that this...
(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an epsilon, N argument to show that this is indeed the correct limit. (b) Use an epsilon, N argument to show that {1/(n^(1/2))} converges to 0. (c) Let k be a positive integer. Use an epsilon, N argument to show that {a/(n^(1/k))} converges to 0. (d) Show that if {Xn} converges to x, then the sequence {Xn^3} converges to x^3. This has to be an epsilon, N argument [Hint: Use the difference...
Proof of the smoothness rule by providing details of the proof that t (n) ∈ Ω(f(n))...
Proof of the smoothness rule by providing details of the proof that t (n) ∈ Ω(f(n)) whenever (n) is a smooth function and t(n) is an eventually nondecreasing function such that t(n) ∈ Θ(f(n) | n is a power of b).
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT