Sage-Tips

What is the solution of recursive equation?

0 = B + C; 2 = A + B + 2C; 3=2A + B + 4C. This is a system of linear equations with the unique solution A = 3, B = 1, C = −1. Therefore the explicit solution to the recursion is an = (3n + 1) − 2n. satisfies the homogeneous recursion.

What is the solution of the recurrence relation an 6an 1 9an 2 with a0 1 and a1 6?

Exercise: Solve the recurrence relation an = 6an-1 − 9an-2, with initial conditions a0 = 1, a1 = 6. Exercise: Solve the recurrence relation an = 6an-1 − 9an-2, with initial conditions a0 = 1, a1 = 6. Solving these equations we get α1 = 1 and α2 = 1. Therefore, an = 3n + n3n.

How do you find the solution of a recurrence relation?

Assuming you see how to factor such a degree 3 (or more) polynomial you can easily find the characteristic roots and as such solve the recurrence relation (the solution would look like an=arn1+brn2+crn3 a n = a r 1 n + b r 2 n + c r 3 n if there were 3 distinct roots).

What is N in a recursive formula?

We saw in Sequences – Basic Information, that sequences can be expressed in various forms. A recursive formula designates the starting term, a1, and the nth term of the sequence, an , as an expression containing the previous term (the term before it), an-1.

Is the sequence an A solution of the recurrence relation an 8an − 1 − 16an − 2 if?

Notice that if an = 2n for all n, then 8an−1 − 16an−2 = 8(2n−1)−16(2n−2)=4·2·2n−1−4·4·2n−2 = 4·2n−4·2n = 0 = an, so this sequence is not a solution to the given recurrence relation.

Which of the following is a solution of the recurrence relation an 6an 1 9an 2?

an = 6an−1 − 9an−2, a0 = 0,a1 = 1. This homogeneous recurrence has characteristic equation x2 − 6x + 9 = 0, which has a double root of x = 3. Hence, the general solution is an = α3n + βn3n.

What is the solution to the recurrence relation an 5an 1 6an 2?

What is the solution to the recurrence relation an=5an-1+6an-2? Answer: d Explanation: When n=1, a1=17a0+30, Now a2=17a1+30*2. By substitution, we get a2=17(17a0+30)+60. Then regrouping the terms, we get a2=1437, where a0=3.

How do you find the recurrence relation for a recursive algorithm?

For recursive algorithms, cost functions are often not homogenous because there is usually a non-recursive cost depending on the input size. Such a recurrence relation is called a linear nonhomogeneous recurrence relation. an = c1an−1 + c2an−2 + ··· + ckan−k which is the associated homogenous recurrence relation.

How do you solve a recursive sequence?

Substitute back up the “chain” using actual values. Solve the recursive sequence f ( x) = 5 ⋅ f ( x + 2) − 3 and f ( 12) = − 4 , calculate f ( 8) .

How do you know if recursion is finite?

If every recursive step shrinks the problem, and the base case lies at the bottom, then the recursion is guaranteed to be finite. A recursive implementation may have more than one base case, or more than one recursive step. For example, the Fibonacci function has two base cases, n=0 and n=1. Recursive methods have a base case and a recursive step.

How do you interpret recursion formulas?

Recursive formulas give us two pieces of information: The first term of the sequence The pattern rule to get any term from the term that comes before it Here is a recursive formula of the sequence along with the interpretation for each part.

What is a recursion step in calculus?

In a recursive step, we compute the result with the help of one or more recursive callsto this same function, but with the inputs somehow reduced in size or complexity, closer to a base case. Consider writing a function to compute factorial. We can define factorial in two different ways: