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Answer:
Step-by-step explanation:
we want to figure out the general term of the following recurrence relation
we are given a linear homogeneous recurrence relation which degree is 2. In order to find the general term ,we need to make it a characteristic equation i.e
the steps for solving a linear homogeneous recurrence relation are as follows:
- Create the characteristic equation by moving every term to the left-hand side, set equal to zero.
- Solve the polynomial by factoring or the quadratic formula.
- Determine the form for each solution: distinct roots, repeated roots, or complex roots.
- Use initial conditions to find coefficients using systems of equations or matrices.
Step-1:Create the characteristic equation
Step-2:Solve the polynomial by factoring
factor the quadratic:
solve for x:
Step-3:Determine the form for each solution
since we've two distinct roots,we'd utilize the following formula:
so substitute the roots we got:
Step-4:Use initial conditions to find coefficients using systems of equations
create the system of equation:
solve the system of equation which yields:
finally substitute:
and we're done!