A recurrence and its general solution

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Before you start

Linear recurrence calculation. Read the relevant method, then decide which parts you can solve without hints.

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Read an original example question

$u_{n+2}=5u_{n+1}-6u_n$ , with $u_0=2,\quad u_1=5$ .

Part a

Find a closed form for $u_{n}$ .

Part b

Hence find $u_{3}$ .

Part c

Verify the closed form in the original recurrence.

Model solution and review criteria

Part a

The characteristic equation λ²−5λ+6=0 has distinct roots 2,3. Thus $u_{n}$ =A2ⁿ+B3ⁿ. A+B=2 and 2A+3B=5 give A=B=1.

Obtain the characteristic equation and both roots.
Use two constants for a second-order recurrence.
Apply both initial values.

Part b

The closed form gives $2^{3}$ + $3^{3}$ =35.

Use the closed form rather than assuming a geometric sequence.

Part c

Each root t satisfies $t^{2}$ =5t−6. Multiplying by $t^{n}$ shows $t^{n}$ obeys the recurrence. Their sum obeys it too, and the initial values agree.

Substitute or use the root identity.
Check both initial conditions.

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