Recursion finds solutions to a recursion or a system of recursions. For example, to solve the recursion y(n + 2) + 3y(n + 1) + 2y(n) = 0, choose Recursion from the Solve submenu.
Solve + Recursion
yn + 2
+3y
n + 1
+2y
n
= 0,
Solution is :y
n
=
2y
0
+ y
1
-1
+
- y
0
- y
1
-2
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You can also specify the initial conditions by listing a system of equations in a column matrix.
Solve + Recursion
, Solution is :
y n + 2
+3y
n + 1
+2y
n
= 0
y 0
= - 2
y 1
= 1
y
n
= - 3
-1
+
-2
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This closed-form solution now makes it easy to find specific terms. For example, if you define y(n) = - 3(- 1)n + (- 2)n, then y(n) can be directly evaluated.
Evaluate
y20
= 1048573
Systems of recursive equations also can be solved.
Solve + Recursion
, Solution is :
y(n + 1) + z(n) = n + 2n+1 z(n + 1) - y(n) = n + 3 - 2n y(1) = 1 z(1) = 2 y
n
= - 1 + 2n, z
n
= n + 1
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In particular, z(10) = 11 and y(10) = 1023.