2. Let an be the number of ways to write the numbers 1, 2, 3, . ...,n in order such that no number i appears immediately before i + 1 (this is a question which appeared on the exam, where an explicit formula was requested!). Explain why an will satisfy the recurrence an = (n − 1)an-1+ (n − 2)an-2 for every n ≥ 2.

Q: how work and explain that the sequence {an} is a solution of the recurrence relation an = an−1…

A: If  is a solution of  then it must satify this.

Q: Solve the recurrence relation an = an-1 +6an-2 for all integers n ≥ 2, with a = 3, a₁ = 3, a₁ = 9.

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Q: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: Find a formula for 1/2+1/4+1/8+.+ 1/2h by examining the values of this expression for small values…

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Q: Suppose you are walking up a 15-step staircase. Each step you take is either 1, 2, or 4 steps at a…

A: Suppose you are walking up a 15-step staircase. Each step you take is either 1,2 or 4 steps at a…

Q: Show that the sequence {an} is a solution of the recurrence relation an = (1 - k) an-1 + kan-2 for…

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Q: Find the solution to the following recurrence: an = 6an-18an-2+ 9n for n ≥ 2 with initial conditions…

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Q: Can anyone help with the following recurrence relation? Struggling to solve the non-homogenous part?…

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Q: Find the general solution of the recurrence relation: a) an = 7an-1-12an -2 + 2n(5¹) b) Show that…

A: a) Arrange as The auxiliary equation is Multiplying by  gives , so (m-3)(m-4) = 0, so m=3 or m=4So…

Q: 8.) Let (an) n zo be a sequence defined by an=n2²+5 Find the first 4 (four) elements of the…

A: We have to find 1st few term of the given sequence.

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: (a) How many three digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, 6 and 7 such that…

A: Any number which has 0 or 5 in the unit place is divisible by 5.  Case 1: when unit digit is 0 The…

Q: Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or…

A: The recurrence relation for the number of ways to climb n stairs if a person climbing the stairs can…

Q: Find a recurrence relation to arrange flags on an n/2 meter tall flagpole if a red flag is 1 meter…

A: Consider the provided question, We have to find the recurrence relation to arrange flags on an n/2…

Q: 8. Use mathematical induction to prove "for every positive integer n, n³ - n is divisible by 3".

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Q: -, A country uses as currency coins with values of 1 peso, 2 pesos, 5 pesos, and 10 pesos and bills…

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Q: Let tn be the number of ways to arrange cars in parking spaces, where the cars are either hummers(…

A: Please check step 2 for the solution.!

Q: Consider the recurrence relation an = 3an-1 + 10an-2 with first two terms ao = 5 and = 6. a₁ = a.…

A: This question related to discrete mathematics.

Q: Show that the sequence {an} is a solution of the recurrence relation an = an−1 + 2an−2 + 2n − 9 ifa)…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: Let an be the sum of the squares of the first n natural numbers. Set up and solve a recurrence for…

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Q: The general solution of the recurrence relation an Зал-1 is An = a, where a is a constant an = a…

A: Option (3) is correct.

Q: 4) Solve the following recurrence relation using generation functions: an = 6 an-1 - 9 an-2, n2 2,…

A: Solution

Q: 5. Show that an = 2" - 5" is a solution to the recurrence relation, an 7an-1-10an-2. What would the…

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Q: Let n be a positive integer. Suppose that there are three pegs and on one of them n rings are…

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Q: Solve the recurrence relation together with the initial conditions give an=an-2 for  n≥2, a0=5,…

A: Solve the recurrence relation together with the initial conditions give an=an-2 for  n≥2, a0=5,…

Q: Solve the recurrence relation: An = 8an/2 + 6n with a2 = 12. Assume n has a power of 2.

A: Given recurrence relation can be written as: An=8An2+6n This relation is of the form: An=aAnb+fn…

Q: Give a recurrence definition for the set of X of all natural numbers that are one or two more than a…

A: We have to give a recursive relation for the set of natural numbers that are one or two more…

Q: Which of the following is the solution of the recurrence relations below an = an -1+5, ao = - 2 {an}…

A: First option is correct. an= 5n-2

Q: Find and solve a recurrence relation for the number of ways to park motorcycles and cars in a row of…

A: Let an be the number of ways in which we can park the vehicles on (n+1) spaces.Since  we want to use…

Q: 4. Consider the recurrence relation az = az-1 + 2k, k 2 1 and a, = 3. a) Write the first 5 terms of…

A: The detailed solution is as follows below:

Q: a. Prove that (2) is divisible by n + 1. Hint: Simplify (²2) - (2+1).

A: see solution below

Q: Suppose that there are n = 2k teams in an eliminationtournament, where there are n∕2 games in the…

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Q: Solve the following recurrence relation by iteration: an = an-1 + m, n2 2, a1 = p, m + 0. Show your…

A: We will find out the required solution.

Q: For the sequence 2, 9, 16, 23, 30, 37, 44, 51... find (a) closed formula (b) recurrence relation and…

A:

Q: Suppose that there are n = 2k teams in an eliminationtournament, where there are n∕2 games in the…

A: Here, n=2k