3) (a) Find a recurrence relation for the number of ways to climb n stairs if stairs can be climbed two or three at a time. (b) What are the initial conditions? (c) ow many ways are there to climb eight stairs?

13. Solve the recurrence relation. Given:a0=3 a1= 6 an= 6a n-1 + 7a n-2 Show your work in the space provided. a. Find c1 and c2  an= (6)a n-1 + (7)a n-2 c1=  c2=    b. Substitute c1 and c2 into the following equation:  r^2- c1r - c2= 0  show your work here:     c. Identify a, b, and c in the quadratic equation. a=  b=  c=  d. Use the quadratic formula to find the two roots. Here is the quadratic formula: -b + Vb^2- 4ac/2a Show your work here: e. Substitute two roots, r1 and r2 into the equation an= a1r1^n+ a2r2^n Show your work here:     f. Now substitute to find two equations, a0 and a1 Remember to use the equation you found from step e. show your work here: a0= 3 =a1= 6 = g. Add the two equations together find a1 and a2  a1=  a2=   h. What is the solution to the recurrence relations? an=    i. Find the 10" term of the sequence, using the solution to the recurrence relation you just found. Show your work here:  a10=

Determine if the sequence {b„} is a solution to the recurrence relation an = 2an-1 + 3an-2, Vn > 2. %3D a) bn = 3n+1, Vn > 0 b) bn = n+1, Vn > 0 %3D %3D

Give a recurrence relation that would fit the sequence a1, az, A3, A4, A5, .= 1,2, 5, 26, 677 ...