converting a recursive sequence to explicit sequence

I'm happy to tell you that there's an easier way. We could explicitly figure out what they are by finding the eigenvectors of A, constructing the matrices P and P^-1, multiplying those three 2 x 2 matrices with the 2 x 1 vector on the right, and actually extracting the expression for Q(n) from that. Step 1: Identify the nth term (an) of an arithmetic sequence and the common difference, d, Step 2: Put the values in the formula, an+1 an + d to find the (n+. To get an explicit definition, we need to make the sequence above fit a quadratic function: At this point, you've probably been told to create a system of three equations using f (1) 5, f (2) 10, and f (3) 17 in order to solve for a, b, and c. Converting recursive & explicit forms of arithmetic sequences. Find the common ratio by dividing any term. These can help: CC.4 Evaluate recursive formulas for sequences CC.4 Evaluate recursive formulas for sequences - Algebra 2 QB9 CC.6 Write a formula for an arithmetic sequence CC.6 Write a formula for an arithmetic sequence - Algebra 2 H82 CC.7 Write a formula for a geometric sequence CC.7 Write a formula for a geometric sequence - Algebra 2 Q5V. Similar idea for arithmetic sequences, only its multiplying a common ratio here for. This tells us that there's some invertible 2 x 2 matrix P (consisting of the eigenvectors of A) such that: = P ^n P^-1 Īnd so: = P P^-1 įrom this we easily deduce that for some constants a and b: Q(n) = a(1/3^n) + b(3^n) How To: Given the first several terms of a geometric sequence, write its recursive formula.

The eigenvalues of that 2x2 matrix are the roots of its characteristic polynomial: det(A - xI) = (10/3 - x)(0 - x) - (1)(-1) = (x - 1/3)(x - 3) Let's call that 2 x 2 matrix in the middle that we're raising to the nth power, A.Now how do we quickly compute powers of square matrices? When they're diagonalizable, it's easy. Instances of Point, LineString, and LinearRing have as their most important attribute a finite sequence of coordinates that determines their interior. We can apply the same logic to our matrix equation to obtain: = ^n Arithmetic Sequences Definition, Explicit & Recursive Formulas.

This is the 2-dimensional analogue of a recurrence like v(n+1) = a*v(n) which has an easy solution v(n) = a^n * v(0). That second equation seems silly, but it allows us to write the following matrix equation: =

Converting a recursive sequence to explicit sequence