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9. Prove that the following formula holds for the kth-order differences of a sequence ho, h1,... , hn, .. . A k hntj j=...
(a) Find the first four terms of the sequence of heights h0= h1= h2= h3= (b) Find an explicit formula for the nth term of the sequence Suppose a ball is thrown upward to a height of ho meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hn be the height after the nth bounce. Consider the following values of ho and r. Complete parts (a) and (b) below. ho = 35, r=0.75
using J-K flip-flops, design a synchronous counter to produce the following repeating sequence 0,6,2,4,0 and prove it in Multisim.
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves asymptotically like ー2+1 where j is the largest order of all diagonal submatrices J of the Jordan form with o(%)-ρ(A) and v is a positive constant. (Hint: refer to Greenbaum for an expression of the kth power of a j-by-j Jordan block)
Suppose I want to prove that a property P holds for every natural number Quasion 9 Not yot arewaed Pi holds for every natural number i amaler than k, then PA) holds Now I show that every natural number k, Points but at This means that P hoids for every natural number F Fag gesson Select one True Falso Queion 10 Suppose I want to prove that a property Pholds for avery nonnegadve intogor. Now I show that for every...
Therom 1.8.2 n choose k = (n choose n-k) n choose k = (n-1 choose K) + (n-1 choose K-1) 2n = summation of (n choose i ) please use the induction method (a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
Problem 3. (20 pts) (a) (10 pts) Show that the following identity in Pascal's Triangle holds: , Vn E N k 0 (b) (10 pts) Prove the following formula, called the Hockey-Stick Identity n+ k n+m+1 Yn, n є N with m < n k-0 Hint: If you want a combinatorial proof, consider the combinatorial problem of choosing a subset of (m + 1)-elements from a set of (n + m + 1)-elements.
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...
2. Use induction to prove that the following identity holds for al k 2 (n 1)2"+12 Be sure to clearly state your induction hypothesis, and state whether you're using weak induction or strong induction
Write a formula for the nth term of the following geometric sequence '3 9' 27 Find a formula for the nth term of the geometric sequence. n- 1
(1) Consider the following processes: There are No = 1 many individuals in the zeroth generation. The number of individuals N in the kth generation comes from each individual in the (k-1)th generation having Poisson(A) many offspring independent of all others. (a) Find a formula for E(Nk). (b) Suppose X1. Show that P(Nk 0) converges to unity as ko N. = 0) converg (2) Consider the processes from the previous problem modified so that the number of offspring which each...