Give a recursive formula s(n) for the sequence of squares [1,4,9, 16, 25,...] of the form...
Give a recursive formula t(n) for the sequence of central binomial coefficients (m), [,2,6, 20,70,...] of the form where a and b are real numbers and Cn is the nth Catalan number. You do not need to prove that your formula is correct.
A sidewalk with n squares (in one long row) is to be painted. Each square will be painted red, blue, or yellow with the property that adjacent squares are always colored differently. Let on be a sequence counting the number of ways to color a sidewalk of length n. (a) Compute c1, C2, c3, and c4. (b) Find a recursive formula for cn (c) Find a closed formula for cn (d) Use induction to prove that your closed formula is...
Give a recursive formula for the function g(n) that counts the number of ternary strings of length n that do not contain 2002 as a substring. You do not need to find a closed form solution for g(n).
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F for all n E N and with Fi = F2= 1. to find a recursive relation for the sequence of ratios (a) Use the recursive relation for (F) Fn+ Fn an Hint: Divide by Fn+1 N (b) Show by induction that an 1 for all n (c) Given that the limit l = lim,0 an exists (so you do not need to prove that...
Start with xo -11 and use the recursive formula: Zn+1-2 Tn in order to compute r2. Give your final answer in the form a/b where a and b whole numbers Start with Ro-7 and use the recursive formula: ^n+l,2 Tn in order to compute r4. Give your final answer in the form a/b where a and b whole numbers Use the Newton-Raphson method in order to find the recursive formula for estimating the cube root of a. Use the formula...
I need to create fibonacci's formula in C++ using a recursive and non-recursive formula. After doing so, I have to time each operation for for the following numbers of the sequence: 1, 5, 10, 15, 20, 25 I have to time how long the program took to perform each number recursive vs non recursive. Code in C++
I would appreciate any help on this problem for discrete math. Thanks! (: 15. (Q1, P4) Consider the sequence of partial sums of squares of Fibonacci numbers Just to check that we're all on the same page, this sequence starts 1, 2, 6, 15,40, (a) Guess a formula for the nth partial sum, in terms of Fibonacci numbers. (Hint: Write each term as a product.) (b) Prove your formula is correct by mathematical induction. (c) Explain what this problem has...
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive definition of I(s), the length of a string s E Σ For a bitstring s, let O(s) and I(s) be number of zeroes and ones, respectively, that occur in s. So for example if s = 01001, then 0(s)...
25. Find the first five terms of the recursive sequence an = n + 3an-1 where a = -2.
Problem 1: Give the exact and asymptotic formula for the number f(n) of letters “A” printed by Algo- rithm PRINTAs below. Your solution must consist of the following steps: (a) First express f(n) using a summation notation 2 (b) Next, give a closed-form formula for f(n). (c) Finally, give the asymptotic value of the number of A's (using the O-notation.) Include justification for each step. Note: If you need any summation formulas for this problem, you are allowed to look...